Playing with one other dog will not be sufficient for your pup’s socialization. Instead, they should meet, and play with as many other puppies and dogs as possible.
It’s important that your puppy enjoys being restrained and examined. It’s easy to train them to enjoy it with daily cuddle sessions filled with lots of handling and treats.
Dogs at play will often look and sound like they are fighting. With a well-trained dog you can always check that things are in control by interrupting the play session with a bit of obedience.
Letting the little dogs off-leash first gives them a chance to explore and get comfortable. Some pups are naturally shyer, and need more time to build up their confidence.