Only one person can claim the deduction - you or your child. Your return could be rejected if your child claimed the exemption on their tax return, and didn't check the box that someone could claim them as a dependent.
When I had a similiar situation with my children, we ran our tax returns both ways to determine who got the biggest tax savings (me), and then filed to maximize the tax savings. I then paid them the difference between the taxes owed with and without the exemption, and still benefited some from claiming them as a dependent.
