I am a woman, and in particular, a trans woman. I've got an F on every piece of legal identification I can. My birth certificate is the last one, and requires surgery because of the state I was born in.
Disclaimer: I am speaking only for myself here.
Personally, I just want my legal/medical identity to be respected consistently. That includes locker rooms and other gender/sex-segregated spaces.
When it comes to folks like CDs, at least to me, it's more complicated. I don't particularly want men in women's spaces, regardless of what they're wearing. That said, I have no problem with CDs being in there, because statistically it's just not an issue, and they have safety concerns. My personal belief is that CDs should feel okay with going into the womens room for safety reasons, but I don't believe we need to legislate the right for them to be in there. It should not be a punishable crime, but if asked to leave over a dispute, I don't think it's unreasonable to leave.
The tricky thing here is what constitutes a "sincerely held gender identity", which is language that gets used a lot. For trans men and women who transition, IMO there's very little room to question, especially with legal identity changes. In the other cases, I personally do wish we had some kind of standard, because it would shut a lot of nasty people up. That said, I'm not certain it's possible to define a standard that's actually enforceable, other than "what's on your legal ID", so here we are.
I don't love the fact that my choices appear to be "bathroom free-for-all" and "legislated bigotry that denies men and women access to the correct spaces". If anybody offered me the chance to vote for "guaranteed access is based on federal identification, and using the 'wrong' bathroom is a non-punishable/non-convictable offense in case of dispute", I would do so in a heartbeat. You would then find me advocating FIERCELY for removing (or substantially easing) the monetary roadblocks that can keep some people unable to make those legal changes.