They do not have to report on their tax return, and in fact they did not have to issue a 1099 in the first place. If otherwise qualified they can use the expense for the child care credit assuming the babysitting was to work or go to school

Out of curiosity did they hire a contractor or did they have a household employee?

It was a contractor.  For household employees, I believe they have to issue a W2? Is that correct?

Correct, a household employee is issued a W2