How to configure X for multi-head with two separate screens?

I would like to try out the old-school way of handling multiple monitors with X - where each screen is separate, with a different $DISPLAY value (the first screen is :0.0, the second is :0.1 and so on), and where windows cannot be moved from one screen to the other - as described here.

At the moment, I do not have any X config files for the displays, and I am getting a mirrorred desktop i.e. both screens show the same windows and my mouse. $DISPLAY is currently set to :0

What do I need to do to make X treat each monitor as a separate display?