Kuchi ga katai (口が堅い): the mouth is hard. Not telling others what should not be told. The Daijisen dictionary gives the example kuchi ga katai otoko da kara shin’yō dekiru, he keeps things to himself, so he can be trusted.
Kuchi ga karui (口が軽い): the mouth is light. Talkative, and saying even what should not be said.
Ofureko (オフレコ): off the record. In a press conference or interview, an agreement not to record or publish certain remarks.