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.
