I am going to attempt to answer this even though I'm probably wrong, but I feel like taking a shot at it.
Person is singular, so its possessive form would be person's.
Persons is plural, so its possessive form would be persons'.
However, I think "missing persons" is just the title of the report and isn't meant to show possession. It's just meant to categorize people who are missing. It's not the missing person's (or persons') report, it's a police report. The possession of the report belongs to the police, not the missing people.
Like I said, I'm probably wrong, but I wonder about things like this all the time, like for instance, if it's people's or peoples' since the word people already is plural. I feel like exercising my brain today, so I answered.
What we do in life echoes in eternity.