Are they logged in as application users? Since the user is TRYING To save a user defined version of the report, the application needs to know their user name/id to save their specific version of the report..
To save a public report, the developer must select Save Public Report must be selected under Include in Actions Menu on the Report Attributes page. Developers can also apply authorization scheme to enable or disable Save Public Report. See "Customizing the Interactive Report Search Bar".
So it's possible you don't have this set in your application report setup..