Rather than getting a full-blown Isaac equivalent, sidekicks steal a heart from Player 1's life and spawn as a randomly-generated baby. If Player 2 willingly drops out to give Isaac his health back, they'll play as the same type of baby if they drop back in at a later point, but getting killed off means a fresh spawn from the baby generator, which offers types including Love Baby, Psy Baby and Spider Baby. Player 2 is free to return as a freshly-spawned baby so long as doing so doesn't leave Isaac with less than one heart. Also, so long as Player 1 doesn't slap the controller out of their hands for wasting health.
McMillen stresses that the game is balanced for co-op, so you won't be mowing everything down just because you have double the projectile tears. Babies don't have the same capabilities as Isaac either, so they won't be able to pick up objects, nor will they determine when items are used. Babies also have a "cursed" attribute, like automatically dropping a bomb every minute, a risk McMillen equates to the downsides of the game's pill power-ups.
If you're worried about co-op's addition hindering your ability to showcase your talents, McMillen's FAQ states that there will be some related Achievements, but assures that co-op challenges won't stand between players and the 100% completion Platinum God Achievement.