Many good ideas in this thread, but i think live missing dwarves should operate mostly internally, with overseer reports ocurring only if the missing dorf is stranded/walled in. Some thoughts.
1) I love the idea of having sheriffs/fortress guard be responsible for finding missing dorfs, since missing dorfs are potential crimes, which are their responsibility.
1.1)In fact, since they are the fortress' internal security, it would not be a stretch for sheriff/fortress guard to periodically engage in "patrol" tasks where they actually go around the fortress, keeping tabs on dorfs, asking questions (thus learning who the questioned dorf saw) and maybe even catching crimes red-handed. Thus, they are more likely to know the location of most dorfs and resolve missing reports immediatly. This will also incentive large fortresses to assign more people to the guard if your captain is spending most of its time patrolling a huge fortress alone. (ninja'd by chthonic :p)
2) Regardless of how the missing live dorfs are implemented, i dislike the idea of hiding missing dorfs from the player's view. Missing reports could serve as valuable warnings that a dorf is walled-in/stranded, and hiding the missing dorf only makes his rescue more difficult. At most, make it impossible to zoom in creature the missing dorf on the unit screen, but it should be visible on the main screen and obey orders as normal.
2.1) Basically, this could make the sheriff/fortress guard work as a unit-list analogue to a record keeper. If you ignore it or put too little time/dwarfpower on it, you will have lots of missing reports and lose the precious ability to zoom in on the missing dorfs.
2.2) What i worry about is the announcement spam. All of this missing dorfs systems needs to be balanced to avoid excessive announcements.