Yes, they do! And I agree, they're usually flopping down on my lap right after they do, so I get just as wet.
Here's a picture of Errol's tummy a couple of weeks after getting a haircut. Our groomer actually cuts the hair on their private parts very short so they don't make a mess when they potty. Then there's a circle of hair around that area that is cut to about 1/4" or so. Then it gets a little bit longer as it moves up their chest.
Hope this helps!