Why is it "men" rahate hain? Why is "men" in the sentence? It seems like a literal translation of this would be "Neha and Peter live IN together"?
The official translation is completely correct. "Saath men rehna" is phrase that just means living together. It can apply to children living with their parents or two brother living together or any such thing.
"saath men rehna" has no connotation of a live in relationship whatsoever.