“Each other’s lives” is better because we are using “each other” to refer to the individual parties. Therefore, we must stick to the individual plural form of “lives” when we want to show that there are multiple lives being mentioned. These examples should help you to understand more about it: Correct: I want us to be a part of each other’s lives.

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.” Which is correct: each other’s life or lives?

It all depends on the number of entities involved. If two entities, "each other's life." (There is only one "other life.") If three or more,"each others' lives." (Here think of the missing understood words, "of the" between "each" and "other."