Personal-/Private Message (PM) in the context of instant messaging is so wrong. It indicates that something should be private or personal. Something between you and the recipient. Well is it? Is it really private? Is your communication taking place over an encrypted protocol and is it transient and never persisted? If persisted, is it then encrypted where you are in control of the keys? I think a better wording for this is "Direct message" as that indicates the behaviour of the communication. The message you are dispatching is directed. It's intended for a specific receiver. It says nothing about the privateness.
Just a reflection after hearing "PM" repeatedly just the last few days.