I know that my girls often start squealing with each other as a game.  This can escalate to exactly what you have described.

Try parting on good terms with the other boy.  Like, wave bye bye etc.  Don't make a fuss when they do start in on the crying as you are actually giving them a signal that it is ok for them to be upset like that over nothing, which really is what it is, nothing.  Kids hey.

Maybe you and the other mum could get down with the boys and play ball as a team.  Children learn by example, so if you and your friend have fun playing together then the boys might be encouraged to join in. 

The other thing you could do is create your own sign language so that when you and your friend are trying to talk and they start screaming, stop using words and use your hands instead.  this will capture the boys attention, and they might try and join in.  Worth a try right?

I wonder if he is feeling jelous about the whole thing, both of them.  Have you tried to switch babies?  Holding your friends son and her holding yours?  Do you tend to give more attention to the other baby when you are together, or to your friend, and not much to your son?  It very hard at this age for them to have enemies, they are not even at that age where they are able to interact and play together, developementally.  My guess, with not seeing it, is he may be jelous or threatened in some way.  And threatened, I don't mean like doing to be hurt, but maybe cause he knows you, he knows that the attention from his mommy will be taken away.  You say this is new, it probablly is something that he is now realizing is going on, and just give it time.  But keep at it, and try when you meet together to bring something special for him and your son to play with, and take the first few mins to give only your babies attention, then give the other baby attention, then talk to each other.  Good luck and hope things get better soon.