Looking on interweb, it looks like one unit on the 1000. It's quite possible that when removing it you could put coolant in the engine, but at that stage the engine is probably drained of oil anyway.
On the fzs600, the oil pump is gear driven on the right hand side of the engine behind the clutch.. It has a shaft only that goes through the engine and drives the water pump on the other side. 2 separate components although sharing a shaft.