In Chapter 2, Concepts and Permissions in Reports and Dashboards, we discussed Unfiled Public Reports briefly. This folder is accessible by any user, so any user with Run Report permissions will be able to run the reports stored in this folder, and to copy (save as) the report from this folder to other folders (including to the user's private folder). There is nothing that can be done to secure this folder.
But this does not mean you can delete or modify any reports stored in this folder. You will be only able to delete or modify reports that you create and store in this folder unless you have specific permissions--we will discuss this further in this chapter.