In modern English, people is the de facto plural of person. People and person have separate Latin origins, and they came to English at different times by different paths, but there are examples of people used as a plural of person from as early as the 14th century. Persons was the original plural, and it is possible to find examples of its use in all types of writing up to the present, but it prevails only in a few contexts, most notably law and law enforcement, and in a few common phrases (e.g., persons of interest, displaced persons, missing persons). Elsewhere, it usually gives way to people.
There is an old usage prescription holding that people applies to uncountable groups of individuals (e.g., Times Square was packed with people) while persons applies to groups that are easily counted (e.g., there were four persons on the balcony). But there is no good reason for this distinction, and in any case it is not consistently borne out in real-world usage.
The question I would ask in this situation is, does "people people" sound right?
I personally would say that it doesn't and would go for people persons.