People are always talking about each other behind each other's back.
Should be other's or others' or if they are both ok, and why?
Should be back or backs here?
The effect of each other is confusing me here.
The sentence is grammatically correct. but you are right, it feels clumsy, and the mixture of singular and plural is uncomfortable. A quick check of google suggests both singular and plural used. The plural, "backs", seems more common.
Talking behind someone's back implies talking about each other, so you don't need to say the first part, and we can look for synonyms for the second. How about:
People are always gossiping about each other.
People are always spreading rumours.
People are always speaking ill of each other. (more formal)
There are also some gendered insults that could be used here, but I couldn't recommend them.