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?
You can connect them via bluetooth using BluetoothSockets. Android developer website has pretty good documentation on this.
http://developer.android.com/guide/topics/wireless/bluetooth.html
Or if you'd rather (and have internet on both devices), you can use regular Socket's.
http://developer.android.com/reference/java/net/ServerSocket.html for server side http://developer.android.com/reference/java/net/Socket.html for client side
If you have a large amount of data to transfer, internet sockets have a greater data capacity and will be faster. The other advantage is that there is no such thing as "out of range". You can connect the two devices wherever internet is available, whereas with bluetooth they have to be within bluetooth range of each other