As a member of a Mighty Network, you can send and receive private, direct messages with other members in the same network.
Start a new direct message with a another member by navigating to their profile and tapping on the button that says "Say Hello."
Once you've messaged at least one other member in your network, you'll see a chat icon in the app along the bottom if you're using an iPhone. Tapping on the chat icon will open up your main messaging area, where you can always access previous conversations.
On the web, you can access your direct messages with other members by clicking on the chat icon in the upper right corner.