In the phrase missing person, is the whole phrase a compound noun or would missing be considered an adjective that modifies person? It seems like in many situations when it is used with other adjectives, missing is not treated as an adjective. In addition, I have seen the phrase pluralized as missing persons rather than missing people, which makes it seem like the phrase is viewed as one whole word.
Since one can be on an official list of "missing persons", and it means something slightly different than a list of "missing people" or a list of "people missing", (either of which would probably mean that they were not in an expected place) I'd have to say that "missing person" is a compound noun - as it has a distinct meaning when phrased that way.
The difference between missing persons and missing people would be that missing persons means that the individuals that are considered here as persons are missing. Where it shows individuality. But missing people is more like a group of people missing.
Hence I would say that "Missing person" would not be considered as a phrase and "Missing" here would be considered as an adjective.
The fact that 'misper' is an acknowledged (if slang) shortened form of 'missing person', as shown in Wiktionary
misper English Noun (plural mispers)
(police slang, term of art in private investigations) a missing person
shows that compounding is accepted for this concept. Though the 'rules' can be rather arbitrary;
(5) I want a baked cheesecake not an over baked one.
All the items are baked.
shows that some structures seem intermediate between '[true]adjective + noun: collocate' and 'adjective + noun: open compound'.
This example, and further discussion of the debate, is given in the paper HOW TO UNDERSTAND THE ADJECTIVE PLUS NOUN COMPOUND AND ITS ADJECTIVAL COMPONENT?.
I'm not the best with stress, but it seems like there is only one stress on the phrase "missing person" which means that it is a compound noun. Regardless, it is a frequent phraseology.
Interestingly, "missing person" and "missing persons" are quite frequently used as adjectives. Missing people, however, always seems to be a noun.