Yes. When you enter the childcare expenses you also must enter the Social Security number or Tax ID of the person who was paid to take care of your child while you worked. The IRS will match that up with the income reported on the tax return your sitter files.
It could raise a red flag with the IRS if you claim child care expenses and the child care provider is not claiming the income on their own return.
In order for you to claim the child care credit, you will be asked to provide either a Federal ID number or a Social Security number for the care provider. The IRS may use that information to cross reference the care provider's tax return to be sure that income has been claimed. There is no way to know for certain that will happen, but it could.
The child care provider has a responsibility to properly report their income and you have a right to claim a credit if you are eligible.
How must tax they actually pay depends on their overall tax situation, but when you put their SSN on your tax return to claim the credit, the IRS will be trying to match their tax return to yours.
Also, if you paid someone who worked in your home, they are your household employee and you must give them a W-2 and withhold and pay self-employment tax. If they worked in their home, they are self-employed. You don't give them any paperwork, they are expected to keep their own accurate business records of income and expenses.