I've only had one child who didn't night train when they day trained (all within a few months of turning 2), so I can't figure out why it wouldn't be a good thing to night train too? (Though I'm not sure how you'd do that - mine were just dry at night and we peed first thing in the morning, and if they woke up at night it was b/c they had to pee so we brought them to the bathroom.

But your sitter doesn't need to do it - you can do it for sure. It seems weird she'd offer unless something about your DS peeing in a diaper at night is inconveniencing her.