If I report that I paid child care expenses, to a relative babysitting, for example, do they have to report that income, or is there a threshold before needing to report?

Yes, there are "thresholds" for reporting income and filing a tax return. For 2021, please see Do I need to file a federal return this year?

If they don't file a return this year, the IRS may send them a letter to them requesting information as to why they didn't report the income from you.

State filing requirements may differ. And the reporting rules for the relative are applicable even if you choose not to claim the expense yourself.

Q. And what happens if if I report the expense and they don't report the income?

A.  A red flag will be waived at the IRS.  Depending on the other person's overall income tax status, a notice may or may not be sent.

Depending on the details, that person (or you) may owe FICA (social security and medicare) tax on that income.  FICA has a lower threshold than income tax (as little as $433).