How do I set up IntelliJ on dual monitors?

All 'windows' inside of IntelliJ other than the code editor tabs have a "Float" option which pops open an independent window which you can drag to anywhere on your screen. This will allow you to manage your IDEA windows however you like.

IntelliJ also allow you to open multiple projects at once. When opening a project with one already opened, you are prompted as to wether you would like it in your current window or a "New Instance". If you choose new instance, it will open up a completely new IntelliJ with the other project in it. This will work for you if you have many projects you work on at once.

IntelliJ 14 and onwards you can select the settings cog and tick floating mode to detach menus, or drag editor tabs off the main window to detach them

If you're running on OSX you'll need to make sure it's not treating the screens as separate spaces, otherwise they'll snap back onto the same window as the main editor when switching between windows.

To do this untick Displays have separate Spaces in the Mission Control System Preferences.