You wife's business has the responsibility to issue a Form 1099-NEC to the contractor if your wife's business paid the contractor more than $600. That's where the responsibility of your wife's business ends with respect to the contractor.
It's the responsibility of the contractor to file his own tax return, include the income on Schedules C and SE and pay any taxes due. Schedule SE is where your contractor's tax return will show the calculation of his self-employment tax that covers his Social Security and Medicare taxes which will be included in his balance due.
