Tuition and fees

Whatever education expenses you pay for a student who is not your dependent is considered as paid by the Taxpayer claiming the student. 

You may claim the expenses you paid for the college student who is your dependent and apply for an education credit if applicable. 

According to the IRS:

“Expenses paid by others. Someone other than you, your spouse, or your dependent (such as a relative or former spouse) may make a payment directly to an eligible educational institution to pay for an eligible student's qualified education expenses. In this case, the student is treated as receiving the payment from the other person and, in turn, paying the institution. If you claim the student as a dependent on your tax return, you are considered to have paid the expenses.”

Pub 970

If there is a 529 account established, you can contribute the Room and Board amount into that account and then make a distribution to pay the Room and Board. 

The slight advantage to doing this is that the contribution MIGHT be deductible on your state return. 

In order to claim the tuition credit, the student must be your dependent. So, in your case, you claim the tuition credit on your dependent. The other parent claims the tuition credit on his/her dependent.  You each get to count the tuition paid by the other parent, in calculating your credit.  You also get to count the third third (I assume paid by the student).

Although the general rule, in taxes, is that you must be the one making the payment, to get the deduction or credit, there is an exception for education.

Room & board are not qualified expenses for claiming the tuition credit.