Translation:We received each others' gifts
I think that situation would be very unfortunate. Don't they mean "We received gifts from each other"?
In American English at least "We received each other's gifts" could (and likely would) mean that we each received the gift that the other sent, i.e. we each got the gift intended for us.
If I heard the above, as a native speaker of American English, I would like it was as Catriona28745 thought, that I got the gift meant for the other person and they got mine. I would not think we each received a gift from each other unless "from" were in the sentence.