I've worked in children's services here in NY for 33 years. I'm not a verified expert, so this is all MOO.
It's my experience that the kids who run are those placed as JD/PINS, and they most frequently run to a friend's home (including boy-/girlfriends), or take off with friends and spend a few nights sleeping rough before getting caught and either returned to placement or remanded to detention. It's not very often that they return home and a parent hides them, because at that point they've been involved in the juvenile justice system for long enough that the parent has lost the notion that we are stealing their children and want the kid to return to placement so they can be "fixed". We honestly have had very few situations where a parent has harbored a true runaway.
On occasion we have parents who have lost custody due to alleged abuse/neglect, and have taken off with their kid, convincing the child to leave the foster home to go to the park or stay after school and then secretly leaving with the parent. I don't really view those as runaway situations because the parent has orchestrated things so they can take the child. Surprisingly enough, those situations don't happen here very frequently, either.
When we do have a runaway, the Caseworker files a missing person report with LE and I believe there is a contact within NCMEC we work with directly. We make regular contact with the bio family (both near and extended) and work with various agencies like the school to identify the child's friends or other relationships and other likely places to search. We've been fortunate in that most kids are picked up fairly quickly, but there have been cases where we've had to go out and put up missing posters and do some pretty out-of-the-box things to find them.
I hope that helps. I'm sure that other states and bigger districts have more runaways that my medium-sized district, and there are always those situations that don't fall into "typically" or "usually"
