Does a parent and a child both put in the 1098-T form while doing their taxes?

No, you do not both enter the Form 1098-T on your returns.  Below is some advice from a TurboTax help article:

Parents: If the student listed on the 1098-T is your dependent, enter the 1098-T on your return, even if your dependent paid the tuition

Students: If you're not being claimed as a dependent, enter the 1098-T on your return regardless of who paid the tuition, unless it was your employer. In that case, just keep the 1098-T with your tax records

To learn more:  

Where do I enter Form 1098-T (Tuition Statement)?
 

Q. Does a parent and a child both put in the 1098-T form while doing their taxes?

A. Simple answer: No. 

But, more accurately; usually not.  Only one of you claims the tuition credit.  If, you are your parent's dependent, the parent claims the tuition credit and enters the 1098-T on their return.  If you are not a dependent, you may be able to claim the tuition credit, in which case the student would enter the 1098-T. Be advised; 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.

But, there are times when both the parent and student may have to enter the 1098-T, usually with adjustments. The two most common situations: 1. the parent is claiming the tax credit and the student is reporting taxable scholarship as income and/or 2. the student needs to report expenses against a 529/ESA distribution.