When it's a stranger like someone who comes to my door or that I meet no I don't panic at all.
I do however get that panicky feeling when a Jdub is trying to corner me which happened to me a few months ago via a person who my sister sent to ask questions she won't ask me. That really bugged me and showed me how fearful my own sister is of her religion, half way through my conversation with this person if occurred to me what was going on and I totally relaxed and chose my word very carefully. I guess it worked because after a few months I received a "friendly" email from my sister.
I really hate what this religion does to families. My sister is so full of fear of displeasing Jehovah and her Elder husband to have a friendship with me it's sad, very, very sad. I love my sister and miss her a lot yet I know things between us will always be strained due to her beliefs. Regardless of how immersed she is in this religion she is very weak about standing up to her beliefs and I really feel it's because she knows they are wrong and is very conflicted about them. I used to be the same way so maybe there is hope for her.