Files that are downloaded on the chromebook are saved to that specific device. They aren't in Google or anywhere else in the cloud. They are saved in the Files app on each individual chromebook.
It is a good idea to move important files to Google Drive so you can save them, print them and/or submit them.
To Remove Profiles (Removing a profile will delete ALL screenshots and downloads saved on that device):
Click on the drop down arrow next to the profile/name
Click 'Remove Account'
Click 'Remove Account' again