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?

[teamwork911]

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

[HelenC12]

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.

[SweetieJean]

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

[Hal_Al]

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).