Some do, some don't. Both the cats I have at the moment are dribblers - one will drool at the slightest provocation, the other only if she's enjoying a particularly good cuddle. I don't recall the previous cat we had dribbling at all (but that could be a faulty memory....)
Pixie's always been a bit of a droolpuppy. Whenever she lays on me and is really relaxed she's like a faucet, lol. I asked the vet once and he said it wasn't a problem, she was just very content and relaxed.