> or the school does
The school almost certainly does.
https://www.copyright.gov/circs/circ09.pdf
Tests are explicitly mentioned in the list of works make for hire, but it's also within the scope of employment, so it's double counted.
> If a work is made for hire, the employer...is the initial owner of the copyright unless both parties involved have signed a written agreement to the contrary.
Fair use covers the school's use of students work. Submission to such services might also be construed as fair use. It seems to have been litigated as such already:
https://www.eschoolnews.com/2008/04/17/judge-turnitin-com-do...