Not quite always. While I have no personal experience, I understand that corn dogs served in correctional institutions do not have a stick.
But, at a state fair...... certainly on a stick.
In Japan, you can buy them at convenience stores (they are at least on sticks there), and they are called American Dogs. I do like 'em but don't see 'em too often...guess a Midwestern state fair is in order for next year.