The jet skis I have been on and around have a plug, and they operate like a boat. Don't put plug in: it fills up with water. Put plug in: keeps water out. I took my jet ski to the island many times (as well as dozens of other people) and left it sitting for many hours on end, not once did it ever sink from not running.
It did sink, I was out of state at the time, and a family member decided to take it out. They flipped it upside down, and were more worried about drowning in four feet of water than flipping the jet ski back over. Some nice guy offered to help them out, he towed them and the jet ski to the boat ramp, and must have hit every stump or rock in the bay on the way back.

I got back the next day and tried to revive it, but the engine had already seized. There's $12k down the drain, a brand new Kawasaki 1200STX-R. The worst part of all was that I cancelled insurance the week before, saying "what could possibly go wrong." Famous last words. . .