First of all I never argue over private message, that's a colossal waste of time and inbox space. It's the path people take when they're too scared to make their opinions known on a forum and I'm not going to waste my time supporting that.
Secondly internet arguing is comparable to sparring in front of a listening audience, your only effect on the challenging party should be through an argumentative attack and hopefully good sportsmanship, (don't count on the latter, and don't ever count on somehow convincing your foe of a certain viewpoint).
As of late I've been incorporating a 3 strikes end it rule for my arguments. I end the argument when I feel everything that I had to say has been said already, or when I feel my opponent's argument is no longer worth the time. I still enjoy a nice argument either way.