Dear Kim
Sorry for not replying earlier. In order not to let students see who else has booked the same slot, you need to remove the capability 'mod/scheduler:seeotherstudentsbooking' from the 'student' role. Depending on your setup and experience, a system administrator may need to do that for you.
For general information roles and permissions in Moodle, see
https://docs.moodle.org/28/en/Roles_and_permissions
Best wishes
Henning