It is pretty simple. Either take down someone, or be taken down by an another player to fire up a rivalry beetwen each other. You don't have to take down each other vice-versa or anything like that, either way if two new player destroy each other who haven't met before will automatically ignite a new rivalry. Very important to note: It only works while you are in Freeburn mode. If the host start a challenge, a race or any other game mode, the rivalry won't count - no matter how many times you take down anyone, or how often you taken down by others.