This is not currently possible. By default draw.io files are saved below the "app" folder and these subfolders are not shareable.

If you move that folder to a shareable folder inside Dropbox and open it with draw.io it will state "Note: Your-Diagram will be saved in the app folder" and move it back when it saves.

This is a limitation of the Dropbox app folder architecture, they only allow writing to that one folder and that folder cannot be shared. Only Google Drive implements a reasonable file sharing at the time of writing.