The Kawasaki and Yamaha engines use outside water from the jet propulsion pump. Open loop.

Some of the SeaDoos have a closed loop system with a heat exchanger on the bottom of the hull. These would require an engine driven coolant pump, but I'm not familiar with it.

I guess if you ran an open loop system, you would dump water all over the track and those that ran after you wouldn't get any traction at all!