Question

“On their back” or “on their backs”?

Answer 1

The answer is "on their back." Because you are using a plural pronoun you also need to use a plural noun to agree with it. The people aren't sharing one back.

Answer 2

♥ "Back" Is referring to one person, and since they do not share a back, then it is not going to be "back"
♥ Now since in this case we are referring to more then one person, i can tell by "their"
♥ then were going to have "backs"
♥ Therefore your answer is "Backs"