How can I get notifications to pop up in my active display? I have an MBA running Yosemite. Apps like Mail, messages, calendar, and FaceTime have 'banner' notifications that pop up in my laptop's built-in display but they don't pop up on my external monitor that's plugged in via a thunderbolt adapter. 

How can I get notifications to display on my active display?

Is this even possible or a feature that is yet to be implemented by Apple?
Try this: System Preferences -> Displays and drag menubar (small white stripe at the top of the blue box in the "Arrangement" tab) to your external (attached) display. Built-in notifications will be displayed on the monitor with main menubar.

In addition to this you may wish to explore Growl app (http://growl.info/), which allows you to setup notifications in more sophisticated way and configure them for several displays.