We've long known that The Binding of Isaac: Rebirth will have an all-new local cooperative feature but here are the specifics, as broken down by designer Edmund McMillen.
When the second player presses start, they'll randomly materialize as one of a variety of babies at the cost of a heart container from Isaac. When this happens, "both players' damage/tears are lowered to balance things a bit, but every aspect of whatever items Isaac has at the time of spawning are applied to the baby, including usable items."
When player one uses an item, so does player two. Better watch out! There's also the chance of "special" and "cursed" babies, as well as babies tied to specific characters. (It wouldn't be Binding of Isaac without a bunch of variables.) While player two does have the immediate advantage of flight, they cannot pick up items. They can, however, drop out of Rebirth to return the heart stolen from player one -- and then, if desired, return later as the same baby.
Not your standard co-op setup, that's for sure. "I can honestly say it totally changes the experience," McMillen said. "It's a great addition for anyone who has a friend/significant other to play with and is an awesome way to get new players into the game."