That is an interesting situation and I am sure many others share a similar concern.

I would recommend trying out Podio (free), it lets you create all sorts of templates to capture and work with information (text fields, categories, dates, etc).

You can set it to receive e-mail for you and it will automatically insert the appropriate information into the fields (including attachments). You can decide if your staff only sees the name of the student and the attachment if you wish.

You can also create a workflow that informs you when your staff members finish the corrections and are ready to be sent back.

You can even use it to keep track of all pending essays, their progress and due dates so you are always on time.

Let me know if you have any questions!

Well, it's certainly completely possible to build a system that would do what you're looking for, but I'd like to believe you can find something off-the-shelf that would accomplish it, since a custom system would definitely cost you more (e.g. thousands of dollars to get a decent system built by a reputable firm).

I'm guessing that the students are the clients in this setup?
One thing to watch out for is that if the student/client includes their name and/or contact information on their essay, then you (or the system) would ideally sanitize that information out of the essay before sending to the the teacher/employee.

I don't know a system that I'd recommend for this, but I hope that someone else is able to recommend something that you like.

Why would you ever hire someone you don't trust?

A software has risks, apart from interrupting with other software installed in the computer it also gives easy access to hackers and hacking activities. I will suggest you do it legally.
An employer’s ability to monitor emails sent or received on an employee’s personal, web-based account is more complicated. In general, this type of monitoring is more likely to be allowed if the employee is using company equipment and has consented in writing to the employer’s monitoring of all computer use. Employee privacy in electronic communications is a complicated and still developing area of law.
You can read more here: https://www.nolo.com/legal-encyclopedia/email-monitoring-can-employer-read-30088.html
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath