Yes, Sterndrive (I/O)engines are typically cooled by the type water the vessel is in. Some manufacturers do offer a Fresh water system as an option. This system also varies between manufacturers ie: some cool block & heads & manifolds,("Full") and others just the block & heads ("Partial").
Volvo has introduced a system on their sterndrive gas engines that introduces a salt neutralizing solution, used when the engine is "done for the day". Quite effective, and much cheaper than FWC.
Diesels -Inboards & I/Os (except for sailboat auxilliary's which are typically raw water cooled)) are typically FWC.
As stated above these utilize a "Tube Style" or "Bundle" heat exchanger as a water to water radiator. The incoming sea water will typically cool the H. E. , the aftercooler, and the trans oil cooler, and quite possibly the engine oil cooler-but that varys per mfg.
Bigger diesels found in Draggers, Tugs, and larger commercial vessels are typically "Keel Cooled". The boat has a "grid" under the hull, which acts like a heat exchanger. This system is closed, so the engine has coolant on one side, and the KC loop has coolant on the other.