Well, I think, if a transgendered person has finished the surgeries, and looks like a male/female, he/she can use the mens/womans restrooms. Why not? Before, I think, it's easier to use the restroom of the gender, they were born in.
A close friend of mine is a transgender. He was born as a girl, his name was Christine. He always played football with us , and he was just like us. Wearing the same clothes than us, talking like us, just the same, except, he had no penis, but he always said, someday it will grow on him as well. We called him simply Chris, and he was one of us.
As we were teens, he suddenly became a females body, and he was very unhappy about that.
And for us it was strange, a females body and him, that just didn't match.
Then I moved away, and I lost contact to him.
Two years ago I saw him again, but first, I didn't know, that it was him. A man came to me, and said:" Hey! Do you still know me? It's me, Chris!"
He's got a beard, heavy muscles, a deep voice, there's nothing left of the females body, he once had.
He even got a penis and testicles.
So, why not? He is very happy now, he is married to a beautiful woman, and they adopted 2 kids.
I think, for us, who we are not transgendered, it's nearly impossible, to imagine, what they're going through, and why. I once tried to imagine, what it would be like, to wake up in the morning, and having a females body, everyone would refer to me as a woman. I didn't like this imagination at all.
Face your fears, live your dreams! (No Fear)