I think the first of these that I saw were from Chechnya. Squads would video their ambushes and then the videos would appear on the web eventually. There were some of the attacks into the neighboring republics as well. I think there might have been some older Afghan-Russian ones as well.
Anyway, the NYT has an article about the proliferation of these videos onto youtube and other video sites. The most famous of the sniper videos (right hand photo on the NYT article) has a related story in the Army Times. The medic who was hit in the video (unharmed due to body armor) treats the wounded sniper a little while later.