No, you don't "have to" claim them. But they don't get to claim that $1000 (each) either. So, just claim them and get your $1000 (two $500 "other dependent" credits).
There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working*. He cannot be supporting himself on parental support, 529 plans or student loans & grants. It is usually best if the parent claims that credit. If the parents aren't eligible, nobody gets it.
They cannot claim a credit if they are, or can be, claimed as a dependent by someone else.
If the student actually has a tax liability, there is a provision to allow him to claim a non-refundable tuition credit. But, then the parent must forgo claiming the student as a dependent, and the $500 other dependent credit. The student must still indicate that he can be claimed as a dependent, on his return.
*When the student indicates, in TurboTax (TT), that he cannot be claimed as a dependent, and TT sees his age and student status, an additional support question, about earned income, is added to the interview. TT appeared to allow the student the credit, because you did not go thru the whole interview again (there is not a glitch in TT).
