What is the best way for an Android app installed on two devices to communicate with each other? Can the devices connect directly without using text messaging?
Depends on what you are doing. If you have a server, you may be able to send some message to it and have it pulled by the other device (assuming both clients have the app installed). I think this would be the most intuitive way (but it really depends on what you are communicating).
Text messaging and email might work, but you (or the user) needs to know the numbers/emails associated with a device to do that.