I know a phrase "to speak out of turn" = to say something you should not have said, or to say something for which you have no authority. I've not met "speak out of school", but googling shows it to have approximately the same meaning.
Um... I've understood it to mean - sort of - "I've learned something that shouldn't be talked about except with the people who have already discussed it, but I'm going to tell you anyway."
Engish idiom 
Linear Mode
