1 Each other and one another mean the same.
Mary and I write to each other/one another every day.
They sat without looking at each other/one another.
2 There is a possessive each other’s/one another’s.
We often borrow each other’s clothes.
They stood looking into one another’s eyes.
Each other/one another are not used as subjects.
We must each listen care fully to what the other says.
(NOT… We must listen carefully to what each other say.)
3 Note the difference between each other/one another and ourselves /yourselves / themselves. Compare:
They were looking at each other. A ← B each other
(= Each person was looking at the other.)
They were looking at themselves. A → A themselves
B ← B
(= Each person was looking at him- or herself.)