Generally speaking, not much more than a few bumps and bruises. Even when they may appear to be heavily injured, they can be quickly healed either with the potions or such you can purchase at the stores here, or at a Pokémon Center. Not to mention, they can shrug off attacks that could potentially kill an ordinary human to begin with...

*he shrugs*

I would recommend seeing how interested or not your partner is in battling; Pokémon who want to do so will often try to engage whether their trainer's willing or not, which can be more risky than a properly supervised battle.