Currently, the journals are monitored by a group of Aurors chosen for their discretion and loyalty (not merely to the Lord Protector, but to Lucius and Barty's faction.) They have a list of individuals whose private messages they are not to read, and Draco, Harry, and Hydra are on it. Also Narcissa Malfoy. There are a few others.

However, under the circumstances I think it's highly unlikely that Lucius will be able to keep people out of Draco's messages, even the ones to Harry. Just how compromising are these private messages? Are we talking about minor disloyalty, questioning things you're not supposed to question, genuine sedition, treason...?