They do (usually) use a stone to remove burrs after sharpening, but not every sharpener does because it dulls the blades a little. In fact, if you have difficulty skating on newly-sharpened blades, "stoning" the edges keeps them from feeling "too sharp." I've heard this called "stoning down the edges."
However, the term "stoning" also means to hone the edge and make it a little sharper. It doesn't last very long, but it's an effective technique when there's a skating event for which you don't want too-sharp blades. Instead of stoning the actual edge, you run the stone along the sides, which makes the edges more sharp. This used to be called "stoning up the edges."