You would not report your child's income reported on a Form 1099-MISC on your tax return. That would only be the case if you chose to report their income on your return as opposed to the child reporting in on their return, and the child's income was solely from interest and dividends, with other requirements.
If your child had only self-employment income under $400 or earned income under $15,750 they would not be required to file a return.
You will have to determine the amount and kind of income being reported on the Form 1099-MISC to see if your child needs to file a tax return. You can learn more in this TurboTax article.