How do they deal with the corrosive effects of salt water? Or do they remove most of the salt before using?

Every large ship engine is cooled by raw ocean water, mostly through a heat exchanger system. Basically instead of cooling the engine coolant with air like on your car, they cool it with ocean water.

In my experience in smaller boats using the same system (100-150 feet) corrosion is less of a problem than growth and calcification. Mostly we just dissolve everything with acid every once in a while on those systems.

And sacrificial zinc anodes that get replaced regularly.

True. That covers galvanic corrosion, but not other corrosive effects.

Probably just using stainless steel or similar material for the pipes. It would be too energy intensive to desalinate the cooling water. It is an open system that pumps water in and back out again. It is not a closed system.