According to the IRS, parents who are not married and alternate claiming of a child dependent, "The parent who claimed their child on their 2019 return may have received an additional Economic Impact Payment for their qualifying child. When the parent who did not receive an additional payment files their 2020 tax return next year, they may be able to claim up to an additional $500 per-child amount on that return if they qualify to claim the child as their qualifying child for 2020."
This only references the one stimulus payment but would apply to both.
This is referenced in the last paragraph under Claimed dependents are not eligible for an additional $500 payment in this article: Why the economic impact payment amount could be different than anticipated
This will all be done by claiming the Rebate Recovery Credit on your 2020 return.
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The law was written to allow this, whether from sloppiness or on purpose, I suppose no one can say.
You do need to make sure you claim the dependent correctly. If you live together and share custody, the one who claims the child should answer "no" to the question about custody agreements, because this only relates to a legal custody agreement between divorced or separated parents. The other parent should not list the child on their return at all, to prevent confusion.
If you live separately from the other parent and share custody, you can only claim the child if the child lived in your home more than half the nights of the year; or if the child lived with the other parent more than half the nights, they need to give you a form 8332 release.
But if you manage to follow all those other rules, the child can qualify you for a stimulus credit this year and the other parent last year, and will not cause a problem for the other parent.