Luis
I think the current approach is still solid. While it might be annoying for the person who reported the issue to have their report remain in non-done state for a while before you or Vlad get around to marking it as done, I think there is value in keeping this in the hands of Admins.
I could see there being cases where a bug is only half-fixed, but no longer affects the original reporter so they mark their issue done, and move on — potentially leaving a stone unturned.
The case of duplicate issues is also very minor in my opinion. My understanding is that when a post is flagged as dupe, merging the two issues is a very quick procedure.
Perhaps adding a flag-as-done feature could be a decent compromise, and it might lower the bar for some people to take action (tagging an admin just to tell them that something has been implemented might seem intimidating).
While the thought of the feedback forums being self-moderated sounds attractive at first, I don't think that should be the case and the potential issues, in my opinion, outweigh the pros.