The 1099-Q gets reported on the tax return of the named beneficiary whose SSN is on the 1099-Q. Period.
In a case where the parent is the named beneficiary and that parent has two kids in college, the parent can claim the funds between all eligible students as desired.
Any left over 1099-Q funds not used for qualified expenses (which includes room and board for a student) is taxable income to the named beneficiary on the 1099-Q. Excess withdrawal penalties may or may not apply also.
