I havent had this problem with my dogs, since my older dogs usually put younger dogs in their place. However, with my servals I have had to vigorously train them not to bite me, or my dogs when playing. I'm sure it would work on dogs as well.
Always have a stuffed toy that your puppy enjoys playing with on hand. At playtime, this proves to be really useful. When you play with the puppy, don't let the toy out of your hand. And if it bites your hand, say No and let it bite the toy, and when it does, praise it. You don't really have to treat the dog at all, a "good dog" and a nice pet on the head and back usually work fine. Hopefully, it will get the idea, and you can do this when you see the puppy biting on the dog as well. Good luck with your mouthy baby.
-Calyx