Hi Hannah -
There are two ways to control who can see the responses for a questionnaire.
The first control is through capabilities. You can control capabilities per role, per user and per instance in the standard Moodle way. I have attached a screenshot of the specific capabilities that control response availability for questionnaire below:
The "Read all responses any time" capability is required to be able to view all of the responses regardless of other settings. The "Read own responses" is required to even see your own. If you want to ensure that a certain user or role can never see any of the responses, make sure they do not have any of the response capabilities assigned. For the case above, the "student" role is only able to see their own response.
The other way to control response access is through the specific questionnaire's settings. These options are as follows:
That allows you to control when a student who can access responses is allowed to see them.
From what you are asking, I believe you want to use capability restrictions.
mike