Not much different than you would do on the internet. The Server has to run some kind of Web-Server that can accept and process requests that you send to it. So instead of sending the AJAX Request to a certain homepage you prepare your Webpage (or whatever you want) to send it to the IP-Adress of that Server.
Is there something specific you want to know about?
Edit: Re-read your post. Do you mean to send an AJAX response to the client without a foregoing request of that client?
Yes, without the client requesting it. Basically the client should change a number shown on a screen by a full-screen webbrowser, whenever it recieves a specified AJAX call with a value in it.
I intend to do some handshaking first, obviously. So the client connects to the server, and keeps the connection open, receiving AJAX calls at random, while the client continues to update the number on the screen to the last AJAX call received.