Yes, it is possible to communicate between Android devices or applications that are not connected to the Internet.
If you are using different devices, you probably want to use sockets, but it is possible to run an HTTP server on the server device and then use HTTP. Make sure they are connected to the same wifi network.
There are several different ways to communicate between different processes on the same Android device. The easiest might be to use a Messenger service but sockets and HTTP are also possible here.