Did you scratch each others’ eyes out, or scratch each other’s eyes out? Hard to see the difference? Well it would be with no eyes. It’s actually pretty straight forward.
Each other is a pronoun and is used in the place of a person’s name, in the same way as he, she or they.
A lot of people mistakenly think that because by definition there has to be more than one person involved in each other, it is a plural pronoun, like they, so in possessive form it behaves like a plural noun would – each others’. They put an apostrophe after the s, like they would for a plural noun – employees’, students’ or members’.
But in fact each other is a singular pronoun, it is one entity and so behaves like most singular nouns do in the possessive form, with ’s to show the action, object or behaviour belongs to each other.
So in fact, if you’re doing it right, you’re scratching each other’s eyes out. Remember that, next time you get caught up in fisticuffs.