I am not claiming my children as dependents -- their mothers are claiming them. Why do I still need their SSN to file?

The IRS tracks dependents, and their parents, to make certian that more than one individual is not claiming them

If you are not claiming any child related credits or benefits (claiming the child as a dependent or a nondependent), there is likely no reason to include the child's Social Security number in the tax software or in the tax return.