Do I have to file form 1098-T with IRS? If my child is a dependent on my tax return, does it mean that only one of us has to file the form 1098-T?

Yes, that's correct. If you are claiming your child as a dependent on your tax return...then only you claim the education credit or deduction. You'll enter the information on the 1098-T on your tax return as it applies to your dependent.