The answer depends on whether “others” refers to the singular or plural possessive. If you are using each other (singular), then it is “each other’s.” If you are using the plural, then the apostrophe goes after the “s.”

Yes, “each other’s” has an apostrophe before the “s”

So, like most singular nouns that indicate possession with apostrophe-S, so too does “other”.

We will look at the other people’s houses. Here, “other people” becomes the plural form, and we add the apostrophe and “S” after that fact.