My last cat Simeon used to get abcesses in his mouth because of a bad tooth, which made the whole side of his face balloon like you describe, (he was 18 when they started).
The first time it happened his eyeball actually started to protrude. His face went from normal to hugely swollen in less than a day.
The vet lanced it under his eye (can't tell you how discusting that was) and gave him strong pain killers and antibiotics and he was back to normal in a day or two.
He was in pain that first time, you could tell when he tried to eat, and he couldn't close his eye it protruded so much, which made his eye dry out. The problem recurred every three months until he died just over a year later (of other problems). After the first ocassion, at the first sing of the absess returning he had to go and get it lanced again, but we were able to catch it before it got too big or caused him signs of pain.
If he had been younger or fitter the vet would simply have operated to removedthe tooth and absess completely.