Collisions:
What a player sees on their screen is what happens. If you collide with something/someone, you take damage. If you do not collide, you do not take damage. So basically, if your game engine sees that your plane touched another plane, you will take damage. If the other plane's game engine (the other guy's screen) doesn't see that his plane touched your plane, he won't take damage.
Gunnery:
What the "shooter" sees is what happens. If you shoot at someone and your game engine registers a hit, that info is sent to the other player's game engine and he sees/hears/feels the hits. You are shooting at what is on your screen, not "the clone that's flying around somewhere near the master". Because of the delay of the Internet, it's very possible that someone will get hit (and even blow up) even though on their screen the other plane never had a guns solution.