If a piece of gum is 10 calories, does that include just the chewing or if you swallow it, too?

Chewing. Doesn't matter if you swallow it or not, from a calorie perspective.

Just chewing it cause all the sugar comes out 

It will be almost all in the chewing because the sugar and flavor is dissolved. All that is left is the gum. By the time you've chewed all the flavor out there aren't any colories left in the gum. By the way if you keep chewing the gum for one hour that uses up 10 calories anyway.

So for calorie neutral gum chewing keep it up for one hour.