Many thanks for such a helpful and prompt reply! I didn't know that there was a time lag where making information private was concerned, so that's helpful to know.
I still regard myself as a novice re. family history research, so can appreciate that it might be a "clueless" person behind this, and we've all got to learn from our mistakes (I've certainly been led astray by records at times). The person concerned has posted family photos in his Facebook messages so he probably does think there's a genuine connection, but I just wanted to be sure that I wasn't putting my family at risk.
Re. Facebook: you can control what others see on your timeline/home page, and can change settings for posts, photo albums, etc, if you don't want information to be available to everyone. This contact was made in a private message, however, which I felt obliged to open and read. The person concerned had been searching for me after looking at a surname-type genealogy site and Facebook seemed his best way of finding me (well, it worked!). I'm going to take things very carefully.
Thanks again for your help.