What exactly is Rivals mode? Is it an actual head to head race, or is it more of a leaderboard system like in Trials HD?
The latter.
Basically there's a bunch of set races (number of laps, car, etc). You're given a time to beat which is typically the next person above you on the scoreboard (or the next fastest in your club if you're in one).
When you enter the race you're racing against that person's ghost and you get a bounty prize of credits for beating their time.