The customs of your planet are strange to me.
Ditto, and still do. I’m actually quite saddened that there are grown, intelligent adults out there who not only feel that the only value in a male-female relationship is sex, but who also believe that the majority of adults feel that way. Sure isn’t that way in my social group, and never has been.
Since I posed the question in the OP, I shall put forth my own answer (incomplete as it is).
First off, whether it’s “less horny” or “more picky” or “less imperative & immediate” or whatever, women in this world have some of it even if less than men do. So the reason men tend to come on to women, making overtly sexual overtures, so much more often than vice versa, is not that women are that uninterested but rather that they don’t have to.
If you’re a moderately horny female, sex can happen for you often enough to satisfy you if you respond favorably to some subset of the males who come on to you. If the males completely stopped taking sexual initiative, that would no longer be true, you wouldn’t be getting at all, so perhaps as a moderately horny female you would come on to men as often as you needed to in order to get laid as much as you wanted to.
Similarly, if you’re a fairly picky female, sex with a sufficiently satisfying partner can happen for you often enough if you cull from among the many men making overt sexual initiates those few who meet your picky standards. If, once again, the males completely stopped taking sexual initiative, that would, once again, no longer be true, so in order to couple up with a sufficiently satisfying partner that meets your fairly picky standards, you’d have to choose from among men who weren’t coming on to you overtly and come on to them as often as needed in order to get what you want.
I’m not sure it works at all the same way if the difference is mostly due to feeling sufficiently safe and secure. Emotionally safe and secure with leaving yourself open to getting your feelings hurt, yeah, but if we’re talking safe from the specter of rape, physical assault, and other related stuff like that, I don’t know as how magically making the males all cautious and in need of feeling safe would make women feel any safer or act any bolder, so let’s bracket that one off.
OK, back to horny and/or picky — let’s say the men are as much so as women, no more & no less. That would imply that men would not (necessarily) stop taking sexual initiative, but they would have no reason to do so any more often than the women did. And if women started taking up the slack in sexual initiative, there would be no reason to assume they would start doing all of it (since the men haven’t stopped). So one sort of assumes there would be a balancing-out, an equilibrium. I sort of assume the asking would be more casual, and the accepting or rejecting more casual as well — I mean, as it stands now, when you’re a male doing the asking, you’re distinctively on the Asking Sex side of a divide, and she who is being asked stands distinctively on the Always Being Asked side, but with the horniness or pickiness evened out, presumably you would not have that, so even if you’re a male doing the asking you would not be Of the Asking Sex, your sex would be getting asked overtly and often, so it’s not like you have to ask and succeed or you ain’t gettin’ none. That’s going to affect how you feel about yourself and the process of asking, right? I think it would be like “Hmm, you’re very nice, you know, I’d kind of like to do you / no? / well, too bad <smile and shrug>”. Not so often with so much determined intensity. Not so often with such anger. Probably a lot less hostility towards women in general, that whole “Goddam hot bitches I’d like to fuck them all, chicks acting so Miss Don’t Touch Me like they’re better than me, she’d like it though I fuckin’ assure you of that” sexual-hostility rant that we’ve all overheard a few zillion times.
How about her? When she’s being asked, that whole meat-market Always Being Asked thing wouldn’t normally be part of her personal past, especially if the asking is more casual-like as described above. Would there be insulting aspects of guys far less often being far away from a <shrug>, rarely mounting much of a campaign about it? Then there’s the asking part of it. Maybe that would feel fine, liberating, empowering — pick a selection-worthy guy when you’re in the mood and chat him up and make an overt pass at him. Most of them will have heard it before, more or less as often as you’ve heard it from guys, will that mean he’s less awkward about saying either “yes” or “no” so the whole thing feels less weird, for you to be doing the asking? Or would it feel, I don’t know, demeaning or like coming down to a less exalted level, as if you had to be desperate to be doing the asking like that? To these things I don’t know the answer, never having been female and having no access to current female experience, so it’s really hard for me to imagine how such a situation would make me feel if I had.
Now the safety thing seems complicated. If women were doing the asking as often as the men, then, like I surmised above, it seems like there’s be a lot less of that male hostility and the pushy emphatic way so many men have of asking women for sex. And that would reduce the extent to which women felt unsafe about sex (especially casual sex with folks who are more or less strangers, if that’s the biggest disincentive for women w/regards to casual-cute-stranger sex). But can you get there from here, or is this circular reasoning? The only way women would be doing the asking as often as the men would be if they felt safe and unthreatened, which would transpire, by my logic, if the women were doing the asking as often as the men. And on some base physical level, women being smaller and genital architecture being what it is, you could argue that it’s hard to posit a world where men are as concerned about safety and security in sex as women are. But if we do it hypothetically as an exercise to satisfy the terms of the OP… OK, if somehow all threat of rape and physical assault by males were just completely erased, or at least erased to the point that women felt no more at risk from men than you’d expect men to feel from women, what then?
For me as a male, a blessed relief from female eyes cast backwards not in curiosity or interest but in self-protective assessment of me as a possible danger…relief from female hostility that’s an expression of fear and of anger at being made afraid. Possibly…probably…a world of more receptive female strangers, less of a pattern of female strangers being standoffish and remote in public lest some stranger-guy get “the wrong idea” – ?? But really most of the changes wrought by that would be experienced first by the women, it’s their world that would change principally and most vividly. OK, women, how would that be for you, what would change for you? And even if it’s kinda far down the list of things it would mean for you ;), how do you think it would change your sexual behavior? Flirting, courting, making passes, casual sex yea or nay and/or under what circumstances, etc?
Why? At present, there are men who have sex with men because that’s their natural inclination, and there are men who have sex with men because they are in settings (prison, work camps, at sea, etc.) where they have no access to women and their own hands just don’t satisfy. If men were less horny, the latter group might be satisfied with masturbation, or with nothing; we would still have gay men, but none of them would ever even have heard of a “glory hole.” (Ya gotta be really horny to use a glory hole!)
AHunter3, I think you underestimate how much of our behavior is genetic. No matter how much men are changed, I doubt women will change much. Also :
Probably not; instead, we’d have a resurgence of good old fashioned bigotry. People tend to hate that which is different; with a lowered sex drive I suspect women will be classed with gays by most men. I think you really underestimate how much suppressed hostility there is between the sexes, especially given how repressed it is on the male side.
To make it clear, I personally don’t feel that way towards women, but I think you’re wrong about how many men have little or no interest in women beyond sex. I’d be happy to be wrong, but I don’t think I am.
I will now attempt to say something substantive. 
I suspect with regards to my relationship-initiation patterns very little would change; I have consistently (not quite exclusively, but consistently) been the initiator of relationships I’ve been in. This is primarily because the men I’m attracted to tend to be geeks, and geeks are frequently oblivious; my favorite response to an, “I’m interested in you, treat this data accordingly” was my boyfriend’s, which was (more or less direct quote), “Actually, it hadn’t occurred to me to think about it, but now that you mention it, I’m interested in you too.”
I suspect I might not be actively hostile to propositions. My primary experience with propositions is that they are followed up with pushiness and may well culminate in assault; as a result, I tend to find them extremely threatening. If I were limited in my sexual experience to selecting from people who sought me out, I would not have sex; people who seek me out scare me too much. I consider this an overreaction on my part, but I don’t know how to fix it without extensive therapy; I would be more motivated to seek out that therapy if I had difficulty finding partners because of it, but I currently have three sexual relationships.
Casual sex? I have no idea. I have a notion that in a different timeline (without the aforementioned bad experience set) I might be more liberal in my partner selection and willingness to have sex than I am at the moment, but in this one, the one time I started up a friends-with-benefits relationship it lasted a grand total of a week (which included no actual sex due to external constraints) before turning into a committed partnership. I don’t think I have the data to speculate.
In private life? I think that would depend on what differences there were in specific. As constituted at the moment my sex drive is fairly strongly reactive; I don’t know how much of that is a constraint induced by bad experiences and how much is just how I am natively. If, however, my partners were also shifted to strongly reactive sexualities, I suspect that the sexual activity would go down due to lack of initiation. (I note that as it stands at the moment, I have a higher sex drive than one of my partners and lower than the other two.) If the levels were similar, but people’s proactive/reactive stuff was kept the same (and I know women who are strongly sexually proactive, so that is not restricted by the parameters of the situation), then not much change.
I typically encounter the existence of these people when reading snark/cattiness forums where other people are laughing at them for being utterly pathetic. Not my subculture; I don’t much feel like swapping subcultures in order to find this sort of thing, so I don’t know which ones it would be common in.
Personally, I would probably have had sex earlier and possibly more often. When I was young it occurred to me that if I really wanted sex, all I had to do was walk down the street propositioning men until one said yes. I didn’t imagine it would take very long. For that reason I never felt the need to prove to myself that I could get sex the way perhaps men do.
You’re right that the latter would go down, but a proportionately greater (not absolutely greater, sorry I misimplied that) would be having gay sex, since 1) heterosexual men would be less inclined to “hunt” women and 2) hunting is much less a part of gay pairings. In other words, the gay “magnet” would be weaker but there still would be very few barriers to those magnets sticking together.