I'm very much with leema on this issue. He's a Border Collie for starters, and it is partly because of their natural tendencies towards nipping that they are used so successfully for herding sheep; they nip at the ankles of the sheep. When many Border Collies see moving objects (including children, cars, bikes, birds, bees, etc. etc. etc.), they naturally want to nip at it.
Despite the breed traits, I think you have to play the numbers a little here and ask which is the more likely. Is it that your four month old puppy is 'dominant' and feels it needs to prove this to any dog it is around, or, is it that your puppy is only four months of age and is not yet fully able to control its emotions and behaviour.
Without a shadow of a doubt, my money would be on the latter. If your dog, and other dogs, aren't responding to your pup's biting, I think it's likely because they see it completely different to how you see it. You may see it as aggression, but chances are that they see it for what it is - puppy over-exuberance.
I think it could be counter-productive to believe you have a dominant dog. Doing so may mean that you are inclined to behave and react completely differently to how you may need to be doing. If you get it wrong with a Border Collie, you are going to have far more problems than you might have with many other breeds. Border Collies can be extremely sensitive, and you forget this at your peril. Discipline methods which might, for example, make a German Shepherd Dog look at you funny, can so easily make a BC run for cover, and it can take days for the dog even to come near you again, let alone want to train with you. Just don't risk it unless you are 100% certain, would be my advice.
All that said, you can't allow the nipping to go on unchecked. If you do then you effectively teach the puppy that it's fine, and it will continue to get worse, in terms of the severity of the nip, and what it nips. There are several things that you could try, but because of the sensitivities of the breed, and the age of your pup, I'm reluctant to suggest any. I think you do need a qualified behaviourist (or someone who has plenty of experience with Border Collies) to come along and show you how to deal with the problem, based upon your own pup's personality.
On a final note, don't be in a hurry to get your dogs mixing with every other dog it sees. At that age, I was very strict in that the only dogs my pup ever mixed with was other dogs in the training class, which I knew was controlled. Yes, you want your pup to mix with other dogs, but you also want those encounters to be positive ones. Negative encounters now may cost you dearly in pup's later life. As a trainer friend of mine once said to me: "All I see most people doing when mixing their pups with other dogs is teaching their dogs how to become better fighters."