Why is 대화를 주고받는다 an awkward expression?

 ANSWER: Because 대화 by itself already implies 주고받는다. So, it would be better to simply say 대화한다.

The Sino-Korean word 대화 (對話) means "conversation" or "dialog," and 주고받다 is a pure Korean word that means "give and take" or "exchange." You do not "exchange a conversation"; you "have a conversation."

If you want to use the verb 주고받다 to refer to a conversation, then you should use an object noun similar to 의견 (意見), which means "an opinion," "an idea," or "a view." Then, the phrase 의견을 주고받는다 could translate as "exchange ideas or opinions."