I tend to be like "it's fine I'm paying" and they're like "But you paid for it last time, and I told you I'd pay for the next time" and I'm like "I don't remember that so whatever" and then I end up giving my card and then the other person is like "but..."
So I pay like 3 and they end up paying like 1 at some point later because I'm weird like that, but normally I'd say people ought to want to pay for their own half, which ends up in sharing/taking turns.
Whoever did the invitation is usually the best way, but this generally means the guy is almost always paying and first. In an ideal world, it should split from start to finish, but that won't happen CONSISTENTLY anytime soon.
Normally if the girl likes the guy she will at least offer to split the bill to make a good impression. That said most men are ready to pay the bill if need be.
If a woman doesn't pay her share then it lets the guy know the woman expects us to fulfill our gender role, so she had better be willing to do the same, or cough up her share of the bill. It is the mixing and matching of gender roles that cause so many men to lose respect for women. Pick one or the other because we aren't going to be the only ones stuck in a gender role.
I would like to pay for myself on the first dates because if it doesn't work out I would feel like I have to repay him or something, but if we're in a relationship, I like things the traditional way haha :p