Do I need to report 1098-T that my 18 year old son received, or does he?

Actually, it's more complicated than that.

You report it if you are claiming a tuition credit, but he reports it if he is is claiming taxable scholarship.

Or nobody reports it. The 1098-T is only any informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or deduction or possibly your student has taxable scholarship income. You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T. 

If you claim the tuition credit, you do need to report that you got one (the TurboTax interview will handle this).

That said, there is a tax “loophole” available. The student reports all his scholarship, up to the amount needed to claim the American opportunity credit, as income on his return. That way, the parents  (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship.  You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.

Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 2. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.