“Each” as a subject is always singular: think of it as equivalent to “every one.” The verb whose subject it is must also be singular. Some uses, like “To keep them from fighting, each dog has been given its own bowl,” cause no problem. No one is tempted to say “have been given.” But when a prepositional phrase with a plural object intervenes between subject and verb, we are likely to be misled into saying things like “Each of the children have to memorize their own locker combinations.” The subject is “each,” not “children.”
The tendency to avoid specifying gender by using “their” adds pressure toward plurality; but the correct version of this sentence is “Each of the children has to memorize his or her own locker combination.” One can avoid the entire problem by pluralizing throughout: “All the children have to memorize their own locker combinations.” In many uses, however, “each” is not the subject, as in “We each have our own favorite flavor of ice cream,” which is correct because “we,” not “each,” is the subject of the verb “have.”
“Each other” cannot be a subject, so the question of verb number does not arise; but the number of the possessive creates a problem for some writers. “They gazed into each other’s eyes” is correct and “each others’” is incorrect because “each other” is singular. Reword to “each gazed into the other’s eyes” to see the logic behind this rule. “Each other” is always two distinct words separated by a space although it functions grammatically as a sort of compound word.