If you plan to claim a childcare credit because you paid a babysitter so you could work, then yes, you need to tell the babysitter. The babysitter has to enter that income on her own tax return.
Only childcare expenses incurred so that you could work can be entered for the childcare credit---hiring a sitter for a "date night" etc. does not count.
You will need either the Social Security number or Tax ID of your childcare provider. The amount the childcare provider enters on their own tax return will be matched up with the amount you claim that you paid to them.
https://ttlc.intuit.com/questions/1900643-what-is-the-child-and-dependent-care-credit