I haven't used oil on stones in many years, but what I remember as being "messy" from my previous sharpening is: You normally want the oil to coat the stones, so most people use a finger or two to spread it around. Now you have oil on your fingers. If you're holding the stone in hand, you can get more on your hand holding the stone. I suppose if you were very light on the oil you wouldn't have any overflow. The blade itself tends to collect the oil as you're sharpening it, so it has to be wiped onto something; usually a rag of some sort. When you're done, you need to get the surface oil off of the stone and have some way to store it without oil coating whatever you store it in or near. I usually used a rag to wipe my fingers, the blade, and the stone. I used this same rag to wrap the stone when not in use. To me this was overall quite messy as I ended up with hands than needed a thorough washing, an oily rag, and the same oily rag wrapped around an oily stone.
To me, water, while there's usually more of it, is decidedly less messy. Just wipe it up and air dry the rag. No mess, nothing left behind.
Just my experience and my opinion.
Brian.