Institute beta groups. Groups can be things like Office, Internet, File system, programming, etc.
Find volunteers to assign to a group, such as the Office group. The people assigned to this group should focus on testing office-related apps, such as the Contacts app or the Libre Office suite. These folks can report any bug they find but their primary focus is the apps that fall within their group. Have them use the app on a day-to-day basis and report what they find.
If you open the Contacts app that ships with Ubuntu and click the "New" button in the top left corner, you'll see that the text input box for the Address field is missing. I feel this issue would have been resolved prior to release had someone been focusing on Office apps during beta testing. If a large issue such as this cannot be resolved before release then that app is removed from the release, or replaced with a similar app.
I don't think I've beta tested anything since Dapper (Ubuntu 6.06) so this idea may have been implemented already without my knowledge. Has this type of thing been implemented? If not, can it be implemented? I would definitely be willing to volunteer to be a member of a group.
The reason I don't do beta testing anymore is because I once looked for a bug report, didn't find one, so I tested an app. Upon finding bugs I attempted to report them only to find that there were several reports made just prior to mine. I feel beta groups would help avoid duplicate work/reports.