This is my logic. But I haven't tried anything yet practically. However, I believe it should solves your problem.
1st and easy way is to use jQuery. What you need to do is add a checkbox or a text link or element and than by default hide entire Answers container (parent container is .qa-a-list)
Now if the use want to see other answer, s/he will click on the checkbox or text link to unhide all answers otherwise he will write own and submit.
The only problem with this solution is it wil always keep hidden everyone's answers including the user.
You can modify answers list method in theme or by plugin layer. And conditionally display other answers if users has already given an answer for that question. This is more rock solid but required little bit php works.
So the user will only see the question if s/he hasn't given any answer yet. Once post an answer on that question, s/he will be able to see all other' answers including his/her.
This is not so difficult to do just need to do some R&D and query. You also may like to use AJAX to display answers as soon as user post his/her.