I have no problem whatsoever with "trans" people but do not think it is optimal for, eg, Ms Manning to be simply labelled a woman. She is clearly not, at least not 100%, female, not in the same holistic psycho-physical way that other females are. For example, I'd like to see her try to use a female changeroom and see if the other females go along quietly with that.
Of course Manning should have the right to her name of choice and to be treated as the person she identifies as. But asserting that one can change gender just by announcing it - even when it is a completely sincere statement - seems far too simplistic to me. I of course understand the argument that inventing some new gender designator could further marginalise an already vulnerable demographic, but by simply avoiding the question I think we've dug ourselves into a bit of a hole.
As another commenter said, names are important. It is important for Chelsea to be addressed as she wishes. But it's also important for our language to have a reasonable set of tools to unambiguously describe things. Simply declaring "she's a girl now, and if you disagree with that in the slightest you're a transphobic cis majority oppressor patriarchal bigot", as I see far too much on Twitter, is not the solution.
It's a complex question and I'm not sure what we can do to solve it. I am glad, however, that the topic is on the agenda - the fact we're even talking about it is surely a step forward.