Okay. So if there are two complete sets of sex organs, then you're going to end up with both sex drives simultaneously. The balls are gonna do their thing and want to sow wild oats in anything willing they can catch, but mostly specimens that look beautiful, active, and healthy. Simultaneously the ovaries are gonna do their thing and want to gather resources from physically strong defenders, higher social status members, and the wealthy, offering reproductive opportunities as a commitment to lock in the partner's contribution. You're going to have two separate kinds of horny that each want to be satisfied. So you might be blueballed but not clamjammed, or the reverse. And these two different kinds of horny both pull mostly the same physiological levers even though the mental experience of it is wildly different. So you can get people who are double-horny and simultaneously desperate to be taken as a woman and to deliver as a man. And whether your horny is of the wanting to be thrown down and taken or wanting to jam it in kind, there's still only one physiological lever to pull, the lever that both makes your panties soaked and your skirt tent up in front. It won't blend cleanly mentally, because the gonads aren't coordinating with each other that way. All they can do is give you a double dose of the horny drug that sends all your blood and fluids down south.
Because the act of pregnancy renders the individual vulnerable, nearly all hermaphrodites in nature have some kind of ritual wrestling to determine who "gets" to be the "male" and who "has" to be the "female". The one who isn't strong enough to defend themselves pays the price of raising the offspring, while the one who is stronger (hopefully) takes on the commitment of providing and defending. I expect that this role vanishes immediately once the offspring are born. Once the child is born, nobody is going to care which one of you the baby came out of. No instinct would compel a mother to try to remain "female" in future encounters, that would be counterproductive biologically. They would instead be a better judge of who might be in their weight class to win a mating wrestle with. Meanwhile the "male" might look for a slightly stronger opponent that they can still beat. There's a large evolutionary payoff for selecting the mate as fit as possible but which you can still beat.
I don't see a two-tiered class system forming between impregnators and the impregnated. I see it becoming more like a ranking ladder where every individual mostly gets sorted into their place based on how feminine or masculine they are. Those who go to the gym and work out a lot to be able to win mating wrestles will rise to the top of the male side of the ladder, those who focus on their physical attractiveness and display vulnerability will attract more potential mates and be able to choose the ones with most resources to provide, and as you go through your life you will likely progress from being a more feminine individual supported by the masculine partners, to someone who has started making your own wealth and is slowly losing their looks to age and dating both up and down, to someone with the wealth and power to provide for a younger feminine mate. Women start with their desirable youth and beauty, and that fades over time. Men start with nothing and build their empires until they can retire. This hermaphroditic society would show this kind of transition as an individual ages and becomes less feminine and more masculine. And all the usual cougar / gold-digger / simp / playboy activities can still happen as normal. Belle Delfuta still teases on OnlyFaps that you might be able to top and stick it in if you just send a big enough superchat. Jeanine Stiffler is still gonna Ara Ara Paula Finch. Stacy's Mom's still got it going on, and you still wish you had Jessie's girl. TATU is now about cross-class underage "lesbian" romance with implied risky sexual behaviors pointed the wrong ways down the economic ladder to form the taboo. Gender would be relative. You're a girl to this person who's stronger and richer, and a boy to this one prettier and younger. And how wide that envelope is and how much overlap there is would be akin to how 'straight' or 'homosexual' someone is in our current gendered society. You might find some "gay" strong wealthy masculine person who exclusively wants to be penetrated by some feminine young beautiful person. And like hell the younger person desperate to get their rocks off is gonna say no to that. It's not a perfect analogue, but it's a good example.
I expect for it to be acceptable for any futanari to have a single boyfriend and a single girlfriend without it being considered cheating, due to the dual sex drives wanting to mate in opposite directions along the hierarchy. I expect for this to form chains of wealth distribution passing from the richest most masculine people to their girlfriends, for those girlfriends to be enriched by that wealth to the point of becoming a desirable boyfriend themselves, use some of that wealth they get from their boyfriends to entertain girlfriends of their own, and so on down the economic ladder. I expect this because the two sex drives would be independent, and so getting to blow your load into your girlfriend regularly wouldn't stop the itch to get railed yourself, and letting someone win the mating wrestle will probably just kill the mood. But there will be exceptions to that of course, this is just how I expect things to go in general. In more open societies cheating might not be cared about at all, and paternity tests just track genealogies with no particular social emphasis placed on parenthood. In more fundamentalist societies, masculine individuals probably buy young feminine individuals as wives until that wife can accumulate enough power to buy their freedom and start being a husband to the wives they buy.
Rape will probably be less of an issue, or even a non-issue, as it won't have any sexist component to it anymore. I don't mean "not an issue" as "it won't happen", I expect it to happen more. I just expect people to care less. Everyone is a hermaphrodite, the mating wrestle is ancient biology, and the bidirectional ranking of having people above and below you will make it so non-consensual sexual encounters probably end up with a huffy feminine person pouting and calling the masculine person a jerk and slapping them like a cat in heat after being mated, then storming off and avoiding them or getting backup from a bigger stronger person -- but not a full on traumatic dilemma and criminal prosecution. Something more akin to bullying or mocking and insults, and maybe halfway to just being tsundere about it. Probably frowned on about as much as bullying, and with about as much done about it. Men who get unexpected or unwanted sex are usually not treated as any kind of victim and usually congratulated on getting some, by both men and women. And since everyone here is a hermaphrodite, that attitude is probably the one that's going to prevail. There's no designated "class" of "women" to band together in outrage to form a taboo, no designated "class" of "protector" to label as a perpetrator or to unite together to lynch someone for taking what's theirs, just a matter of what side you are on to who, and for how long. It'll just be a mean thing to repay in kind perhaps. You rape some cute "girl", and then her "boyfriend" shows up to make you "his" "bitch". At least so long as it doesn't involve unwanted pregnancy. That's when you've actually done the harm in this scenario. How society treats that will depend on how communal the childrearing is. If you aren't personally being a parent to your kids, then it's only nine months of annoyance and a fair but low physical risk. Probably treated like rearending someone in traffic. You've gotta pay the bill to fix the car, the court case takes forever, you have to make arrangements for transportation, it costs a lot of money, it's uncomfortable and hurts and you wish it didn't happen, but nearly always there's no lifelong injury to pay out. Then your kid's in the communal pool, you're done being female, and the jerk who put that kid in you is paying their communal share just the same as you anyways. If childrearing isn't communal, then shit's about to get real and the mom's probably allowed to actually just shoot you dead and take all your stuff.
And despite that permissiveness on rape and cheating and relationship stuff, I expect a very weird turn on sex education. Normally parents don't have to worry too much about their boys and girls hooking up too young. Kids masturbate but it's rare for them to go too far with each other too quickly under our current model. But we've got two problems here now, which is that one child alone in their room has both sexes and unless some mechanism is in place to prevent it either physically or through education, they will end up self-impregnating by schlicking with the jizz after fapping at some point just through innate body curiousity. And left unguided, that will happen as young as they can become fertile. Self-impregnation doesn't make clones, it's instead twice as incestuous as sibling-sibling or parent-child incest. The end result of serial selfcest is an entirely homozygous genotype, which can have terrible consequences, and one strange consequence I'll elaborate on later. So there's gonna have to be some kind of education or chastity belt or something. Perhaps an UltraWestermarck effect that makes people entirely averse to anything that might make them self-impregnate. That's the first thing. The second thing is you're going to always be in a mixed bathing scenario. The line between platonic friend and romantic interest is going to be completely demolished. There is no heteronormative girl clique of all-girl friends, nor one of boys hanging out at a club being bros. Sexual tension is going to be there between anyone even passably attractive, everyone evaluating everyone else as both a potential boyfriend and a girlfriend. There's no "safe half" of the population to group them with. Any time ANY two teens run into a closet horny, it's not going to be just 'two boys will be boys just messing around' or 'all schoolgirls are expected get a lesbian crush and lilies will bloom'. They're gonna get up to things that need stopped. How that ends up happening is a question for their society that I can't speculate too much on. Perhaps at birth they're all reversibly sterilized until they're old enough and can afford to undo it. I expect nudity will be even more taboo, with everyone having separate changing rooms instead of communal gym showers, while simultaneously revealing and arousing clothing will be more tolerated and appreciated. Everyone likes to look as a male, and to show off as a female, and there's not going to be many complaints. So when the cheerleaders and the ballerinas are showing off both cameltoe and prominent bulges under the thin nude-colored spandex, that's acceptable and draws cheers from the crowd and smiles from the performers. But an actual nip slip would be reacted to with shock, horror, and rushes to cover up. It would be about the tease, which is safe and appreciated by all, rather than the exposure to others which is taboo and indicates an inappropriate risk and lack of precautions.
I'm going to agree with gendered verbs from the above commentator, but I'm going to add in gendered nouns and pronouns. When you refer to someone more masculine than you (or if they aren't but you want their dick anyways) you would use "his", and when you refer to someone prettier and vulnerable you would use "her" (unless again you want their dick anyways). Caretaking and social and domestic roles would be feminine verbs, tools to perform those tasks would be feminine nouns, labor and conflict roles would be masculine verbs, tools to perform them would be masculine nouns. Might even get into it with the other parts of speech as well. All of them would indicate the relationship you have to them at your current status rather than some gender absolute which doesn't exist. So you ask your boyfriend to sexa you up, and he pins you down and rails you. And then in the romantic aftercuddle he thanks you for the sexo and due to not having her itch scratched in a while unexpectedly offers to have you sexa her and she'll have your baby this time. And other romantic things.
Now back to the strange thing from the selfcest from before. Any adult, by themselves, has the ability to create offspring unassisted by anyone else. Any crazy can fuck off into the woods all on their own and start their own cult of just themselves and their selfmade kids, and you probably can't make that illegal or police it. This can lead to selfcestuous communities forming where this is considered the norm. If this community is sufficiently Spartan about eliminating any members who show genetic defects, you will end up with a community of identical homozygous clones. All the negative traits that make incest a problem will have been selected out, meaning the genetic component of the incest taboo is now gone entirely for those individuals, who are all clones of each other that breed true with no further consequence of incest. Like the genetically identical standard white lab mice we're all familiar with. That group will suffer in genetic diversity bottlenecks like cheetahs do, but modern technology can probably cope with that. They'll all get sick at once from any disease, but any cure that works for one will work for all, and any disease that doesn't kill them will burn itself out on them immediately and go extinct forever after for lack of new bodies to infect. They can all take organ transplants and blood transfusions from each other. Any progress on any cancer treatment for any of them will work for all of them because their cancers won't be unique like in nonclonal societies. And the first time one of them gets the idea to go Quiverfull about it we're in an r-reproduction vs K-reproduction conflict between clonal societies and nonclonal societies. There are going to be entire races of identical siblings, identical parents, and identical children, filling nations with copies of the same person's body. An entire nation of Sasha Grays, another entire nation of Riley Reids, another of Saaya Iries, Fae Reagans, Katerina Kozlovas. And I don't know what implications that might have for societies that don't do this. Without modern technology, such a society would simply die from the lack of genetic diversity. But with that technology, the clonal society is probably superior and might replace all others. The world might become Just Monika.
Of course, all of this is highly speculative and hinges on a hundred thousand years of cultural and technological development that will butterfly anything we theorize away to places we can't imagine.