Do students need different accounts for each class?
No, students do not need different accounts for each class. A student only needs to have one account to enroll in multiple classes.
If a student with an account would like to enroll in another teacher's class, they must ask that teacher to generate a new enrollment code for them. You can read about that process here. The student then logs in to his/her account and enters their new enrollment code into the search box. They will then be asked to confirm that they want to join the new class.
Upon confirmation, that student will be able to log in and see assignments from either teacher.