I've taken missing person reports from all manner of people--spouses, parents, neighbors, friends. Generally, missing person cases investigated by the police fall into one of these categories:
1. runaway teens and missing children
2. people who may be victims of foul play
3. those who have a condition that places them in danger--Alzheimer's, mental illness, medical condition, etc.
In these situations, the person can be entered into the National Crime Information Center (NCIC) system as a missing person. Should a police officer encounter the person and run a check, a missing person "hit" would come back with information to contact the investigating agency.
A case of a missing adult in which no foul play is suspected and no extenuating condition exists may not be a high priority for the police. Disappearing to begin a new life is not a crime.
In the case of a missing person who was traveling, the report is filed at the jurisdiction the person is missing from, not the destination. The agency receiving the report would probably contact the police at the destination for assistance.
For "missing" children, the circumstances are different in every case. Police agencies have learned from botched investigations that you jump on missing children cases fast and with every resource available. The priority may be marshaling search and rescue personnel to initiate a search of the immediate area, broadcasting AMBER alerts, notifying the media, etc. Supervisory personnel decide what resources shall be called, whether other agencies, the FBI, a CART team (see link) are notified.
This link includes a checklist of what law enforcement does when a child is missing: