There are two mistakes I see so commonly in writing — even in the writing of very good writers! — so I’d like to try to help sort them out. They have to do with making “people” and “each other” possessive.
The tendency is to write “peoples’ ” and “each others’.” For example:
1. Many peoples’ talents are kept hidden due to modesty. (wrong)
2. We glanced at each others’ outfits and laughed to discover we looked like twins. (wrong)
The thinking (I think) in a writer’s mind is that we’re dealing with plurals so we need the typical way of making a plural possessive — which is, most often, adding an apostrophe after an already-present “s.” The mistake, however, comes in treating “people” and “each other” as singular, adding the “s” for the plural, and then adding the apostrophe for the possessive.
Since “people” is already plural (just like “men,” “women,” and “children”), you need to add an “apostrophe-s,” NOT an “s-apostrophe.” The correct form for sentence #1 is “people’s.” (Don’t be fooled by the other, though less common, use of “peoples” as the plural form, indicating several distinct groupings, nationalities, or cultures of people, in which case the “s-apostrophe” (adding just an apostrophe where the “s” is already present) would indeed be correct: The Native American peoples’ cultures are so fascinating.
Now, as for making “each other” possessive: The thinking (I think) is that “each” is singular — and indeed it is. That’s why we say Each of the bats has a crack in it, NOT Each of the bats have a crack in it (nor, as is sometimes seen, “in them”); since “each” is a singular subject, the verb must match and be singular as well. But when “each other” is used, we are actually talking about two individuals, so it’s already a plural. No additional “s” is needed to make it plural before adding the apostrophe to make it possessive. Instead, we add “apostrophe-s” as we would to any plural word not ending in “s” (like, again, “men,” “women,” “children,” which are made possessive as “men’s,” “women’s,” “children’s” — never mind the fact that signs in most department stores leave off the apostrophes).
The corrected sentences then read as follows:
3. Many people’s talents are kept hidden due to modesty. (right)
4. We glanced at each other’s outfits and laughed to discover we looked like twins. (right)
(I’m sure it only adds to the confusion when programs like MS Word show “other’s” as a spelling error. Pay no attention! — the software is wrong!! — doubtless designed by people very talented in matters technical and technological but not in matters grammatical!)
I hope this explanation helps. Let me know if the usage is still not clear or if you have other specific grammar questions I can assist with.