Because we don't have a clear way to tie together a user and a natural person, self-assignment allows the creation of sock-puppets for grading or favorably marking friends. At the same time it allows smaller self-organized cohorts to work within a larger class. (My own experience in MOOCs suggests the latter is helpful and should be encouraged.)
I actually think all the submissions should be visible, with those needing review being highlighted. I don't want to prevent a motivated student from reading everything (not that he is likely to review everything).
It would be important to record the time of the review (since remote content could change after review).
I think some sort of peer review is important to the Academic Volunteers International scheme. It also suggests there might be multiple types of reviewers, possibly categories like: current student, student that has completed the course, tutor, community volunteer.