In a quick look at a very good newspaper the other day I saw both each others' and childrens'.
Let's review: Each other is what it is. It exists outside the realm of singulars and plurals, and so there is no plural -- there is no each others. So the possessive cannot be each others'. The possessive is each other's.
Children is already plural. Childrens makes no sense; just as you don't say kidses, you don't say childrens. So the possessive cannot be childrens'. The possessive is children's.