Is there a way to view the files that a person has downloaded without searching through all the drivers again?

You cannot view any files a person has downloaded since you do not have access to their computer... if you meant instead "can I view the files a person has uploaded to this site"?...

Yes, use the Advanced Search link on the left side of screen, type the person's name that your searching for drivers that person submitted into the Submitter box, then click the Search Database button.