hmmm.. from my experience you don't just reply with 잘자요.... it just feels empty hahah~ ㅏoreans are usually more expressive. So, usually they'd say something like this formal: 네, 너도 잘자요~ informal: 응 너도 잘자~

Saying "잘자요" back is probably not common. Most people would at least add 네 as in "네, 잘자요". There can be other responses depending on what's on your mind, etc. - 네, (XX 씨도 / 언니도 / 선배도 / etc) 편히 쉬세요. => Ok, you have a good rest too. - 네, (...) 좋은 꿈 꿔요. => Ok, you have a good dream too. - 네, (XX 씨도 / 언니도 / 선배도 / etc). => Ok, you too. - 네, 내일 봐요. => Ok, see you tomorrow. (if that is the case) - 네. => Ok. - 그래요. => Ok. If you don't have anything particular to say, just 네 or 그래요 is also okay.

네.