It was recently brought to my attention that I barely touched on this in the user documentation. I honestly just forgot about making a section for it.
To use this feature, you need one base forum install, we will call it Base. Set this forum up however you wish, and configure it properly. Unless you are using more than one group, you would more than likely not benefit from this feature. Once you have your forum and groups configured to your liking, create a new page or use an existing one. As admin, place the module on the page by: Select Add Existing Module from the control panel. Now choose the page Base is on then select the Base forum module from your choice of options. Click add to place this module on the page. The 'Copy' version of the forum will appear looking exactly like the original Base. Now select Settings from the new module's Solpart actions menu. At the bottom of the Settings area, you will see Forum Settings. Click the + to expand this. Here you will see a drop down list for group selection and a show footer checkbox. Select the group you wish to 'mirror' with this 'Copy' module and click update. This will enable only that group you selected on this 'Copy' page. This retains all module permissions, but remember this can be overriden based on how both pages are configured. Also note that selecting a group will automatically disable the footer in that module regardless of the checkbox being checked or not. The footer is automatically disabled to avoid What's New and Statistics showing for forums which people may not have access too.
There are many use cases for doing this, one example is how it is done here on DotNetNuke.com. We just want to mirror project specific items on the projects home page. This gives us that option without displaying the entire forum collection again. This is helpful as it shows the forums only pertaining to that module and it enhances performance of the initial view.
Another case would be you want a master forum that only admins see and you use this feature to palce specific groups on pages. This is a great way to configure a single module and restrict access to other forums without configuring them to be Private Forums.