Currently, there's a default selection "Publish" which is pre-checked.
I think it would be better, and easier to understand, to have the default be that the snap is published, and a selectable box labeled "Make Private." That's an affirmative choice to be made and much clearer, imho.
(P.S. When trying to make snaps private after the fact, I kept getting thrown javascript errors with phrases like "fatal" in them.)