Students can comment on Class Story posts from their student account if you have student commenting enabled and they have parent permission for their student account (they will not see Class Story posts in their account without parent permission).
Please see this tutorial for how to enable or disable a student's ability to comment on the Class Story from the website.
Teacher's View of Student Comments on the Website:
Teachers will see a student's full name (as they have it listed in class) when a student comments. They will see the parent's full name (as the parent has registered it) along with the name of the student the parent is connected to when a parent comments.
Teachers can also see which parents and students have viewed the post by clicking on the number of views at the bottom of the post.
Deleting Student Comments:
Any teacher connected to the class, any parent connected to the student or the student themselves can delete that student's comment. This tutorial explains how a teacher can delete a student comment.