Not a teacher
I have only ever come across the expression "It goes without saying"
That means you are expected to do something without anybody mentioning it. For example: Tomorrow is your father's birthday. It goes without saying that you have to give him a present. - It is expected that he will receive a present from you without anybody having to tell you to give him one.
So maybe "Doesn't go without saying" only refers to an action you would only perform if somebody asked you to do so. If nobody would mention it, you would not do it.