I am an INTJ (F20) and find that I do the exact same thing. It's a mixture of things for me, so I do not know if it is a trait of our personality type, or a consequence of my upbringing.

Sharing information about myself makes me feel very vulnerable, and I also dislike sharing more than what I know about another person. Balance is everything to me. (My nemesi-turned-best-friend and I often joke that we stay friends because we know too much about each other to not stay on positive terms).

Its almost a defensive measure I take to ensure no one can hurt me without running the risk of the same in return. Mastermind trait, I suppose?

But for me, it could also come back to having had a very rough upbringing and the need to keep my life private was a matter of safety.

TL;DR: I do the same thing, but I can't confirm if it is an INTJ trait or something else.