Reports UI - Users with View access SHOULD NOT see reports generated by other users

I love the Reports UI but my users who have only View access to Reports or any given Report Group SHOULD NOT be able to see the reports generated by other users. The previously generated reports are great, but they are hosted in S3 and don't check against the user roles we have assigned to our users. 

We have a common set of reports that all of our clients use and when they run them on-demand in the old UI, that's all they get--that particular run. Client ABC can now see a version of a report that was run by Client XYZ by clicking on a link. Not good. Making all runs of a report available for all users with access to those reports creates a management headache.