No. Your child or other dependent would report their SSA-1099 on their own return, but only if they make enough income to be required to file (this is uncommon). If Social Security is your dependent's only income, they most likely don't need to file a return.
In cases where both you and your dependent receive Social Security benefits on the same check and the check is made out to you, only report your portion of the benefits on your return.
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