Create a personal/private project named “Private Messaging”. Then create a conversation for the project and name it after the team member you are contacting. On the first conversation comment, @ mention the team member you are private messaging to. BOOM! Instant Private Messaging with Asana.
Repeat this creation of a conversation for each team member that you want to be able to private message. If you or the recipient of one of these conversations @ mentions any other team member in the conversation, then the conversation is no longer a single person to person private message, and I would suggest starting a new conversation as outlined in the beginning of this paragraph to re-establish privacy.
And of course, set/unset the notification options per the right fit for you.