Learners of English (and native speakers alike) sometimes wonder whether they are supposed to write each other’s or each others’ (or even each others) in phrases like “to hold each other’s hand(s)”. Long story short, the correct spelling is the one used in the previous example, i.e. each other’s. Another example:
This is quite logical. The possessive form in English is formed by adding ’s at the end of a noun, unless it is a plural noun, in which case we write just an apostrophe, e.g. “these teachers’ books” (not “these teachers’s books”). This rules out each others, as the possessive apostrophe must be there.
In the case of “each other”, “other” is in the singular because it follows “each”—you wouldn’t say “each teachers” instead of “each teacher”, would you… By adding the possessive ’s, we get the correct form each other’s.
Plural or singular?
What about the noun that follows “each other’s”—are we supposed to use a singular noun (e.g. “each other’s face”) or a plural noun (e.g. “each other’s faces”)?
The answer is: Both forms are common. Since “each other’s” basically means “(mutually) the other person’s”, and we wouldn’t say “the other person’s faces” (unless the other person is two-faced), it makes more sense to say “each other’s face”. Nevertheless, the plural form seems to be more common in modern usage, so it can hardly be considered incorrect. In summary:
We saw each other’s face. (correct, more logical)