First, start up a normal game or continue one you are already playing. Then invite your friend in.
To start the co-op stuntman game, you have to get a silver star in a driving stunt and then press Y. The stunt can be something like driving against oncoming traffic, near misses, jumps, driver bailout as long as you get a silver star it doesn't matter.
The easiest to use is either oncoming traffic or near misses.
Whichever you use will be the first challenge played. Each game is made up of up to 5 challenges but the game will end when one of you gets to 3 challenge wins.
Each time a challenge is won it will alternate who has to start the next one. Just keep driving into traffic and take it in turns to win / lose.
After all 5 challenges, a new game has to be started by pressing Y again.