Form 1098-T - who gets to claim it?

Q.  Should I skip this for the kids as we already claimed it on our taxes?

A. Yes. Since they are your dependents, the 1098-T goes on your return.

There is an exception.  If any of their scholarships become taxable (because they exceed qualified expenses or because  some of  the qualified expenses are shifted to the tax credit or 529 plan exclusion),they may need to enter the 1098-T to calculate the taxable portion of the scholarship (even though the 1098-T is also entered on your return).

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Qualified Tuition Plans  (QTP 529 Plans) Distributions

General Discussion

It’s complicated.

For 529 plans, there is an “owner” (usually the parent), and a “beneficiary” (usually the student dependent). The "recipient" of the distribution can be either the owner or the beneficiary depending on who the money was sent to. When the money goes directly from the Qualified Tuition Plan (QTP) to the school, the student is the "recipient". The distribution will be reported on IRS form 1099-Q. 
The 1099-Q gets reported on the recipient's return.** The recipient's name & SS# will be on the 1099-Q.
Even though the 1099-Q is going on the student's return, the 1098-T should go on the parent's return, so you can claim the education credit. You can do this because he is your dependent.

You can and should claim the tuition credit before claiming the 529 plan earnings exclusion. The educational expenses he claims for the 1099-Q should be reduced by the amount of educational expenses you claim for the credit.
But be aware, you can not double dip. You cannot count the same tuition money, for the tuition credit,  that gets him an exclusion from the taxability of the earnings (interest) on the 529 plan. Since the credit is more generous; use as much of the tuition as is needed for the credit and the rest for the interest exclusion. Another special rule allows you to claim the tuition credit even though it was "his" money that paid the tuition.
In addition, there is another rule that says the 10% penalty is waived if he was unable to cover the 529 plan withdrawal with educational expenses either because he got scholarships or the expenses were used (by him or the parents) to claim the credits. He'll have to pay tax on the earnings, at his lower tax rate (subject to the “kiddie tax”), but not the penalty.

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board)

   -$3000 paid by tax free scholarship***

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q (on the recipient’s return)

Box 1 of the 1099-Q is $5000

Box 2 is $2800

3000/5000=60% of the earnings are tax free; 40% are taxable

40% x 2800= $1120

There is  $1120 of taxable income (on the recipient’s return)

**Alternatively; you can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. You would still have to do the math to see if there were enough expenses left over for you to claim the tuition credit. Again, you cannot double dip!  When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms. But, it will prepare a 1099-Q worksheet for your records, in case of an IRS inquiry.

On form 1099-Q, instructions to the recipient reads: "Nontaxable distributions from CESAs and QTPs are not required to be reported on your income tax return. You must determine the taxability of any distribution." 

***Another alternative is have the student report some of his scholarship as taxable income, to free up some expenses for the 1099-Q and/or tuition credit. Most people come out better having the scholarship taxable before the 529 earnings. 

Yes, you can skip entering the 1098-T into the students return IF there is no taxable income for them to claim relating to the 1098-T. 

If you entered the 1099-Q's and 1098-T's into your TurboTax program so that your TurboTax program could do the math AND your program did not indicate to you that the students need to claim any income, don't bother to enter those forms a second time on the students' returns. 

If there were more scholarships and/or distributions than education expenses, the students might need to claim income. 

If you used expenses towards a credit, which cause the scholarships and/or distribution to become taxable, the students might need to claim income. 

If none of these things occurred on your return, the 1098-T is not relevant to the students' returns. 

The 1098-T and 1099-Q are an informational forms which are used by software to compute taxable income and/or an education credit. 

The Form is not needed nor entered if neither a credit nor tax will be generated. 

Only the person claiming the student gets a credit. 

Only the person who was issued the 1099-Q or received the scholarship pays the tax. 

@Wallrat