For approximately equal mass and density, no. If one planet fills its Roche lobe before the other, it will lose mass through the L1 point until it either shrinks to within its Roche lobe, disintegrates completely, or provides enough material that the other planet grows to fill its Roche lobe. In the final case, the two planets will take a three dimensional figure eight shape .
Contact binaries are already known to exist in the case of stars and asteroids. The one problem with regards to contact binary planets is whether they would be stable over long periods. Asteroids are small enough that a contact binary can be formed by mechanical forces in a collision. Stars, as far as I know, mostly form contact binaries when the primary starts to evolve off the main sequence. For planets, I'd think the most likely formation mechanism would be a double planet losing angular momentum to a moon by tidal acceleration, but unless the moon was ejected entirely, that process would continue until the two planets merged completely.