I found Piwik as a good open source solution for website analytics.
Is it possible to run Piwik and see its features using fake data rather than a live site?
Not only the demo version which is available in the Piwik site. For example create a website locally to my PC using Joomla, install Piwik and create fake data in order to check its features.
Piwik has a plugin called the 'Visitor Generator' which can generate fake data for testing purposes. It is available here, on the Piwik Plugins Marketplace. You can use it to generate 100% fake data or a mix of fake and real data from your own logs. A blog post giving a features overview with sample commands is here, on the Piwik blog.
If you want to just start using it right away without reading the linked documentation, click on 'Administration' in the top menu and then click 'Marketplace' in the left menu as shown here where you will see the Visitor Generator plugin. Next click on 'Install' (top right of the plugin detail) to install it.