Daystrom, many people consider therapy to be intensely private and are uncomfortable sharing that knowledge with others in their lives. If a person isn't concerned with privacy, then usually a different arrangement is agreed upon. But Ts believe it is best to err on the side of caution in order to respect the client.

Puppy, it sounds like you have two concerns that may or may not overlap. You have a fear of how your T thinks about you, possible judgements, maybe shame about your own feelings about your T, curiosity, etc. So thoughts of being around your T--even hypothetically--outside of the closed atmosphere of session are uncomfortable. I think that's very understandable and pretty common. I think the antidote to that is to keep talking about it with her, and somewhat, the passage of time, more experience with her and trust in her.

But I also hear another issue about being visible in general. And this feeling is heightened when in new circumstances with new people--unknowns. Being visible seems to activate fears of being criticized and shamed, humiliated and mocked.
Perhaps you've had such experiences before; or perhaps you have such general feelings for other reasons and are projecting them onto different situations. It doesn't really matter as the feelings are what they are.

I experienced something similar, most strongly in my teens. Just walking down the street to my friend's house was difficult because I had neighbors who often sat on their porch and I knew they'd see me as I walked past. They had never been unkind to me--just the opposite. But because I was so shamed by my parents, those feelings just enveloped me.

Would it be possible for you to try to separate the fear feelings from the anxiety feelings? The fears are the cause, and the anxiety is the response/defense. You seem to understand cognitively that the fears don't really match the facts. So the anxiety "belongs" to something else. Can you talk to your T about where the anxiety might be coming from?