If you're speaking about a large group of people, and you want to reference something they own, which do you use?
"I correct other (people's/peoples') grammar errors."
If you are speaking about a group of people, large or small, you put the apostrophe after "people." You put the apostrophe after the possessor, which in this case is "people."
So the sentence is: I correct other people's grammar mistakes.
However, if you are talking about a nationality of people or of a race of people, "people" here can have a plural possessor: peoples.
You might say.
Of all the native peoples in the new world, the Mayans are believed by many to have had the most advanced culture.
Who were the gods of these peoples? Who were the gods of the other native peoples? Were these peoples' gods one and the same, or did they vary from tribe to tribe?