You ask good questions. Each individual case is different, but generally speaking when a military person goes missing without any known explanation, he or she is declared AWOL/UA (Absent Without Leave or Unauthorized Absense) and an effort is made to locate them. By law, after 30 days pass without any resolution to their absence the missing person is declared a deserter for administrative purposes. This involves a number of procedures and reports - and yes a great deal of anguish and hardship for family members.
Once on "the Deserter List" (and each branch of the service maintains such a list), that person remains so listed until a resolution (if ever) is reached. The missing person and his family receive no more military benefits from that point on.
There have been cases which have been resolved many years after a military person went missing. A few recent ones involved a body being found or identified. In those cases the military cleared the records of the service members gave them funerals with full military honors, and paid the family back benefits and pay.
There were other cases which were resolved when the missing service member returned to the US and military control after having intentionally left/deserted the service many years before. In those cases, they are given an Administrative Discharge hearing and a board decides upon their disposition (usually some sort of discharge from the service).
In some cases when a person goes missing but that there is a reasonable explanation (such as evidence of foul play, a plane crash, boating accident or other situation with no bodies found or identified, etc.) there might be a finding of death rather than a declaration of desertion.
Again, each case is considered individually.
