The only way you'd be aware if someone has claimed your children on their return is if your electronically-filed return is rejected because of it. Mike 9241 above is correct.
You'd receive E-file reject R0000-507-01, since the IRS doesn't allow a dependent to be claimed on more than one return.
Please see the TurboTax Help article “E-file reject R0000-507-01: A dependent on your return has already been claimed (or claimed themselv...” for more information.
See also E-file rejections related to incorrect or duplicated names, birth dates, or Social Security numbers.
