During the Korean War, western and Soviet pilots regularly engaged one another and yes, shot each other down. The North Korean air force was obliterated in major reversals suffered by the communists in the first year of war, and while the Chinese army was doing very well on the ground, their air force wasn't quite yet up to matching the UN air forces.

Stalin, who had been reluctant to start the conflict initially and already had advisors working with the NK air force, authorized combat missions in late 1950 and Soviet pilots began directly engaging American pilots in early 1951. An interesting air war developed in that it was the first widespread air battle between jet fighters. Western pilots held a favorable kill ratio (officially on the order of roughly 8 to 1), but in considering the skewed ratio, one has to remember that many kill rates are questionable and communist fighters tended to be attacking bombers, while western fighters were attacking enemy fighters in defense of the bombers.

This is all from memory as I am in the middle of nowhere Georgia with a sketchy signal this week, but if you check my comment history, I answered a similar question awhile ago with sources.

I know there have always been rumors of Soviet pilots flying combat missions over Vietnam against US pilots, but I am unaware of reputable sources on the issue. Are there flaired users who can cite something that can support or discount those claims?