The database module doesn't have the ability to set apart fields as for students or staff. You can do a few things though to make it work if you aren't able to put in the dataform plugin.
1. Set up all your fields for both student and staff, then when you set up the templates, have a student section and a staff section just as you would on a paper form. The students can fill in their fields then save. The staff can see what they have done and then fill in their part by editing the post and then saving.
2. Or if you just want an approved or unapproved, you can have the database "graded" with a scale and then create a scale with approved and unapproved on it. Staff can go through and use the scale to pass the message.
3. If you also want the staff to have the ability to comment without having to edit the post, you can allow comments, then check permissions to make sure students do not have permission to add comments. Staff could use comments to let students know why, why not their request is approved and requests for more information.
Question: Are you planning on keeping this information private from other students?
Usually if you are you could require approval before it is visible, but if you are going to use editing, this will probably tell the system that the entry is approved since it is saved by the staff. I know that in the past that would happen, but it may not any more. I am not sure if the grading or comments would give the system that "approved" status. We used to get around that by requiring a certain amount of posts (say 30) before they could see other peoples posts, but then saying the students could only post a lower number (say 25). That would keep the posts private between student and staff.