I was both -- although I'm not a terribly social guy I would go get lit up at the bar every now and then, often ending poorly (puking in the alley, ER, etc.) Most of the time, and especially near the end, I was a private drinker though. I would go to bars on Sundays when I ran out of booze (liquor stores here are closed on Sundays), but other than that I would sit in my room and drink. I lived with a couple other guys and I didn't even hang out with them, I just sat in my room.
I relapsed a number of times when I first tried to get sober and didn't tell anyone, so I would inevitably end up hiding away drinking by myself without anyone the wiser, until I finally just stopped caring what other people thought and quit hiding from anyone. I started to get suicidal near the end as well, I just really didn't care about the future.
So I guess I've been on all sides of the spectrum, but I was most consistently and regularly a private drinker. Social drinking did trigger me a bit when I first got sober. As a result I just tended to avoid social events where alcohol was around.
Now I don't worry about it too much at all. I could go to a bar with friends who are drinking and I wouldn't be phased. However, I don't really have any interest in hanging out with drinkers any more, since being the only sober guy in a group of drunks is terribly boring. I probably still wouldn't bother being around it though, even though I feel secure in my sobriety there just isn't any good reason for me to hang out with drinkers.