How do I go about finding out who claimed my child on their income taxes

You don't. This is a consequence of electronic filing. Whoever files first get the claim and prevents anyone else from electronically filing claiming the same dependent.
You now have to file by mail and let the IRS sort it out.

They will not tell you who filed, but will review the returns and determine who was entitled to claim the child. So you should include documentation with your return.

One of the rules for claiming a child is who physically housed him for at least half the year. (1/1/16 - 12/31/16 in this case)