In my own case I started off by thinking: what would I do if I came across somebody drowning, and I thought I could jump in and manage to save them? Unless I thought that I might well drown myself, I would hope I'd jump in, regardless of whether I knew the fellow or not. To me, non-directed donation was sort of like playing lifeguard.
That said, I did not rush into the decision (partly because at the beginning I had some specific reason to worry that one of my close family members would end up needing my kidney). I spoke with a number of doctors I'm friends with, looked at the medical literature myself, and consulted an eminent rabbi. And my transplant center did lots and lots and lots of testing to make sure it was safe for me to donate.
I got my wife's (not totally enthusiastic) permission to do this just before I made the initial contact with the transplant center. I told my kids, however, only just before the surgery, and, still, very very few other people know beyond that.
Be well,
Snoopy