No, the closed loop system you are referring to is the antifreeze which cools the engine. You do not flush this system (unless you have overheating issues, but that is another discussion). You obviously need to ensure that your antifreeze level is maintained to the appropriate level. Seadoo does recommend changing this every so often, but in Texas, your winters are not that extreme. Perhaps every couple years you may want to think about changing it.
Now, your exhaust system is an open loop system, meaning your ski also draws in water from wherever you are riding to cool the exhaust components. In your case, it would suck water into the exhaust manifold and then expel it back out of the ski.
If you are riding in salt water, you will want to flush the exhaust system every time you ride. Even riding in fresh water, Seadoo recommends you flush the system after each ride as well as there could be minerals and/or other small particles that will adhere themselves to the inside of your exhaust system which, over time, will degrade the performance of the cooling. Potentially this could cause problems in the long term.
To do so, at the rear of the ski is a tube where you attach your normal garden hose. You will NEVER want to have water running into the ski without the engine running. You would start your engine, turn the hose on to a moderate level, run the ski for a couple minutes while running your RPMs slightly above idle and occasionally rev'ing the motor up to higher RPMs (~4000-5000). Then turn the hose off. Run the engine another 20-30 seconds while rev'ing it up to blow out the remaining water that is left in the exhaust. Then kill the engine.
That is flushing it.