The Show Past Answers page would have to be redesigned in order to allow it to work for students. As it currently stands, it could be used to view other student's answers as well. (Note the input areas at the top of the page, which can be used to search for other users/problem-sets/problem-numbers, even using wildcards.
The code that creates the Show Past Answers page could be modified to include that panel only when appropriate (and to check carefully that otherwise the current user matches the one requested). But the code needs to be changed before that will work.