I pay for my students' college tuition but 1098Ts are in their names. Can I still get the deduction? If not, how does my student get the deduction when they don't file?

You can only enter the education expenses and be eligible for any education credit if the student is being claimed as a dependent on your tax return.

Yes.

If the student is your dependent, you enter the 1098-T into your TurboTax program if you are eligible for an Education Credit (more expenses paid than tax-free assistance, such as scholarships) in order for the TurboTax program to make the proper calculation. 

look under Deductions & Credits.

If you have any Education Savings Account distributions, (1099-Q), enter that form first. 

If there are more tax-free assistance, such as scholarships, there may be taxable income. In that case, the student would be liable to claim that excess as income. 

If there is excess distributions (1099-Q) the person that was issued the 1099-Q is liable for claiming that income. 

Pub 970

Q. If not, how does my student get the deduction when they don't file?

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, even if you don't qualify as a dependent, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. It is usually best if the parent claims that credit. 

You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.