My guess is that it's not about trusting you, but simply that she doesn't have current patients (as opposed to former patients) on her flist. I am Facebook friends with a few former students, but I would not friend a current student if I got a request to do so.

It's a little weird that she didn't simply copy and paste the letter into an email, or print it out for you, if she wanted to share it with you, though. But she seems a little technologically challenged, judging by the fact that she is unable to find out how to unfriend somebody on Facebook.