In December '12 App.net released a Message API that made it possible to send and receive private messages to others on the service.
As part of this release we created a base framework for a messaging client called Omega to demonstrate the functionality of the Message API. You can get to Omega to view your inbox and send messages by clicking the "Messages" link in the left navigation bar of the page when logged into Alpha.
This will take you to our native private messaging client Omega where you will be able to start conversations with other ADN members. To create a message channel simply click on the "+ NEW MESSAGE" button and begin typing the user(s) you wish to start a private message channel with.
You can have a 1-on-1 chat...
...or add multiple users to have a group chat.
To start the conversation type in your message at the bottom of the page. Omega also supports file uploads as well.
App.net clients in the Directory may or may not have messaging capabilities so check the features list before choosing a client to use.