Most likely you will not be able to find out. There are dozens of threads with the same question in this forum (I moved it here from Parenting).

This has come about with e-filing. When you e-file, the IRS checks the SSNs used on the form to make sure they haven't been filed previously. If they have the return is rejected for e-filing. So what you need to do is file by mail and include documentation of your right to claim your child. The IRS will investigate and take appropriate action. They may never tell you who claimed your child however.

Sorry, but a "lot of process" is inevitable if someone else claims your children on their tax return.

The IRS has to determine WHO has the right to claim the children on their returns, and they will require you to provide the required proof.

Further, the IRS will NOT tell you who claimed your children at any time during the process duento privacy laws.