"Each other" refers to a singular subject, just like "one another" does. Notice "each" and "one" are singular; so are "other" (an other v. others) and "another" (which is like a contraction or compression of "an other").

However, you should point out to students that each other is treated as a singular pronoun and emphasizes two or more separate people. Each other’s is always correct, and each others’ is never correct.

Because we have prefixed “other” with “each” it has to remain in the singular form. While there is a plural form for “other” which is “others,” it differs from the example of “people” since it has no alternative meaning and cannot be pluralized yet again.

However, there is no plural form of “each other,” so “each others” is actually an incorrect term that can’t be used. “Each other’s” is used when you refer to the singular possessive form of “each other,” which we’ll touch more on later.

Think about it like you are referring to “each other” (singular). In that case the word “other’s” is correct. It is like you are saying “each other’s.” With the plural form being others’.