The possessive of each other is each other's; the possessive of one another is one another's.
British Dictionary definitions for each other
Idioms and Phrases with each other
Also, one another. Each one the other, one the other, as in The boys like each other, or The birds were fighting one another over the crumbs. Both of these phrases indicate a reciprocal relationship or action between the subjects preceding (the boys, the birds). Formerly, many authorities held that each other should be confined to a relationship between two subjects only and one another used when there are more than two. Today most do not subscribe to this distinction, which was never strictly observed anyway. [Late 1300s] Also see at each other's throats.