Is there a way that I can determine which user is running what report(s)?

Well, if you wanted to go through the effort, it is theoretically possible to construct a URL formula which can be put into a button that would write out a log record to a table and then run a report.  You could then know who pushed the button and when. But you would have to build those buttons yourself, there is no native logging capability to know when a button was pushed.  I suppose that you can also look at the report list to see when a particular report was last run, that's an option too.