So the cat you have had for years is ok, its this new little one that has breathing probs. It could be either a cat flu, although unlikely if its not all the time. Or because its a stray (is it?) it may be like one of mine from a few years ago. Whenever he ate, he would make these dreadful gulping noises and breath heavily at the same time. I think it was because he was a stray too, and thought food was so rare he would keep one eye out for anything trying to take it away. So he'd eat quickly and make awful noises.
Sounds like this black one may have a similar problem. The kibble may not be chewed properly, as he thinks he has to eat fast and is swallowing it only half chewed, therefore making those noises.
Dont feel bad about scaring it off either, you know what cats are like, once you feed them they hang around for ages and nothing will stop them! Do you know if anyone owns it? Would you be happy to slowly introduce it to your other cat? Maybe they would end up friends! You never know
One other thing, just to make sure it is ok, try to feed it some really small bits of soft meat by hand. Watch how it takes it from you, and whether it gulps it down.
Good luck! And I hope that helps ya. Hate to think it would need a vet.