I used to go to the same church as one of my T's. She only aproached me when it would seem more weird if she didn't but most of the time I would seek her out for a hug before she left (we had OK'd this in a conversation at her office). It was very helpful to me. She was my first T and I was terrified of her. Being at church with her let me see her interact with others. I was able to learn that she was a genuinely caring person and not just being nice to me because I was her client. Some 10 years after working with her we are friends. She still laughs at how I would cower in the corner of the chair in her office completely terrified of her then come bounding up to her 3 days later at church for a hug!
Another T I worked with went to the same grocery store I did. I knew from our first session that we lived in the same area so had already had the "what to do if I run into you" conversation. She said it was up to me and what I was comfortable with. When I did run into her I said hi to her (by name) and she looked right thru me as if I weren't even there. I stoped working with her shortly there after.
I have never run into my current T outside the office but am pretty sure she would approach me, say hi and get a hug... but that is because she knows I would feel hurt if she didn't.
I would ask about the church thing too... Most of the T's I have worked with are active in churches and many had clients that attended their church. Most said it was a bigger deal to the client than it was to them.