Most of the time, people is the correct word to choose as a plural for person. Persons is archaic, and it is safe to avoid using it, except in legal writing, which has its own traditional language. Peoples is only necessary when you refer to distinct ethnic groups (for example, within the same region).

Peoples or People’s or Peoples’ are all pronounced the same way. Because “Peoples” is usually singular, people’s’ is usually correct.

The correct plural possessive form is “people’s.” The rule is to take the plural form of the noun and add an “‘s.” “Peoples'” is incorrect as the plural possessive form of “person.”

The possessive noun peoples’ is correct when you want to talk about something that belongs to more than one cultural or ethnic group. If you want to talk about something that belongs to one specific cultural or ethnic group, you would use the form people’s. For example, “the Spanish people’s opinions” refers to the opinions of just the Spanish.

The answer is that technically speaking, it’s not wrong. But people has become the more common word to use when referring to person in the plural form. In fact, some style guides explicitly state not to use the word persons.