If both are claiming the children, they are both getting a much larger tax refund than they legally should be getting. The ITIN number scam has been costing we the tax payers billions each year. The IRS know it’s going on since they are
sending 100’s of refunds to the same addresses.
http://www.snopes.com/politics/taxes/itin.asp
Actually, the kids SSN must be provided to qualify for either as a dependent on the 1040, or for the EIC credit.
https://www.irs.gov/pub/irs-pdf/f1040.pdf
https://www.irs.gov/pub/irs-pdf/f1040sei.pdf
EIC is an abomination that should go away regardless. Allows refunds for those who don’t pay any tax to begin with.