What would happen if separated parents both claimed the child on their taxes without the other knowing?

As you must give the SS# of the dependent, the IRS computers oick thatup in a milisecond, deny the deduction to all, and assess penalties and interest to all. They will send you a letter with the bill, or explaining why your (now delayed) refund is less than you expect. I suspect it makes the returns subject to further review too.