Context is king. Would you feel uncomfortable around someone who believed in communism? Yes? Well, what if they only believed in it as a voluntary association, like a commune? What if they try to push their beliefs on those around them? On the other hand, what if they keep their beliefs to themselves?
Different contexts generate different reaction.
Do they understand that in the relevant context your beliefs, your nature, your reactions are very different? If not, are they open to understanding that you don't want to interact in those areas where the differences exist? Transgenderism is of course very different from political beliefs but I think that the concept of context and that of setting boundaries holds true.
For me, sharing a locker or locker room or bathroom would be no problem at all. I'd only be concerned if the other person seemed to have some psychological boundary problems - if they wanted to flirt or interact in a way that felt sexual I'd just make a friendly remark that let them know I wasn't interested. If they had some unpleasant attitude, I'd react in the same way I would if someone who was a straight male had an unpleasant attitude.
Carl Jung's concept of the shadow is interesting in this context. If we have a negative feeling or discomfort when a transgendered person is close by (like in the same locker room) even if they are doing absolutely nothing overt, nor attempting to interact with us, then that is what Jung would have called a shadow reaction.
It is something that lives in the shadows of our subconcious that is reacting to what we see - something that we buried away long ago. There is a lot of that in the United States (and Britain) that relates to sex. It doesn't mean that we have buried sexual identity issues, but that we are uncomfortable in a way that is totally unnecessary and implies a weakness that might have been felt when we were young and still becoming a man or a woman. But it is totally unecessary as grown adults.
In other words, if we are in a locker room with someone like Jazz Jennings and she sees our private parts or we see hers, our parts won't fall off, nor will we suddenly 'catch' transgenderism.
Much of Europe has had common bathrooms for men and women for decades. Many places have nude beaches, nude swimming pools, and not for the purpose of titillation or voyerism, but because their culture doesn't have the same degree of prudery that we do.