You cannot. The IRS will not tell you about any other tax payer or if that taxpayer claimed a dependent.
The only way you could even know if your child was claimed by another taxpayer would be to e-file file a tax return and it rejected because the SSN was already used or paper file and you would receive a letter form the IRS telling you that another the payer claimed the same child but they will not identify that taxpayer.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**