In quicksearch ESO appoints one of the players as host. By what mechanism is unknown, you've got no control over it anyway, so it might as well be random. If that "host" has "record game" checked, the game will be recorded otherwise it won't be.
If you check "record game" in the setup screen, a reasonable person would expect to find a recording on his HD after the game is over, but that's obviously not always the case. It's as if ESO needs to play a game of roulette first, to decide if your game will be recorded or not. Frankly, how they managed to come up with such a frustrating and counter-intuitive system is beyond me.