I can really relate to this. I always imagine my T doing "normal" things, and I wonder how she is in situations outside of T. However, I would not want to run into T when she was grocery shopping, etc. That would be weird, and I wouldn't know how to react. I think I would just like to see her grocery shopping, being a normal person, but I would not want to interact with her while she was doing so. Or sometimes I imagine seeing my T on the subway, reading a book or something, and just being another person on a crowded subway train. To me she would be special, but not to any other person on the train.
Also, my T does not self-disclose at all. I have never questioned her about it or asked her personal questions about herself, but I think that this is part of the reason why I want to know so much about her and am so curious. I definitely do not have this problem in RL with my friends, (ie wanting to know more about them), but with T, because she doesn't tell me anything, I have this intense desire to know. Like WikidPissah said, I know that my T is entitled to a private life, which is what makes me feel bad about snooping online and finding out info about her, but on the other hand, I need to know something about her to make sure she is a real person.
Edit: For example, when I found out that she is not married and has no kids and both her parents have passed away, it helped me to share things with her. She doesn't know that I know these things about her, but without them I would not have been able to share as much of myself. I think it is because I can imagine her maybe getting lonely or sad, and so this makes me see her as having vulnerabilities and being not perfect, which in turn helps me to share my vulnerabilities with her.