The only way to find out if you and/or your child has been claimed on another tax return is to file your own return.
If you or someone listed on your return has been claimed, then your tax return will be rejected by the IRS and the reject code will tell you the reason it is being rejected. One reason would be that either you or a dependent has already been claimed on another return.
You will not know or be able to find out on whose return you or your child have been claimed.