If you use the Google Drive integration to allow students to choose files from their Drive to submit assignments through the Drop Box feature, the students still have to make sure the Teacher has permissions to view.  

The integration allows students to easily navigate to, and locate their documents and associate them with a specific assignment so the Teacher can keep all the submissions in one spot.  It cannot force permissions on Students documents.

We suggest that Teachers instruct students do one of the following:
  1. Create the document as a Public document.
  2. When creating the document, if the student decides to keep it private, share the document with the teacher immediately
  3. When choosing the document for submission, choose to submit as a PDF or file type instead of a link
This will make it easier on the Teacher when it comes time to review the submissions.