Of course, one of the things that makes conversation problematic is that the one term "trans-person", is used to encompass everything from self-id only to fully complete surgical transition.
I can't speak for anyone else, but I would be willing to treat full surgical transition as their adopted sex for almost any purpose, including occupying a women's shelter. I think most people would. On the other hand, for someone who is pure self ID and not even doing any sort of body or social modification, i.e. like our bearded twitter person a few pages back, I'm not willing to treat that person as a woman for pretty much any purpose at all.
So, should trans women be allowed in women's shelters? In my opinion, it would be best to leave the decision to the discretion of the shelter operator, even in the case of government funded shelters. The male bodied person should not be "entitled" in any sense to access, but the staff should be able to use their discretion. Just my opinion.