While AT&T and Verizon may not have had a chance to screw Note 2 owners with unrootbale devices—mainly because they've screwed us over by not issuing KitKat updates—it's safe to say that they can in the future. And while Sprint and T-Mobile subscribers have been able to root, the various methods can make the process somewhat confusing, especially for first-timers.
But now, a new root method from renowned hacker Geohot—the same guy that brought iPhone jailbreaks and PS3 hacks—will allow us to root our Note 2s in the easiest way possible, with just a single click. In essence, you need to download and install Geohot's TowelRoot app, run it, and...well, that's it!
Update: Looks like AT&T and Verizon customers will have to wait just a bit longer, as some are reporting that the method is not rooting their devices. The current version of TowelRoot does not root these variants, but the dev is aware and a fix is in the works. That being said, I'd advise you to try the process anyway and report from results in the comments. Worst comes to worst, you won't be rooted, but the method will have no adverse effects on your device.
Note that the video above is showcasing the Galaxy S5, but the steps will be exactly the same for any variant of the Galaxy Note 2, as well as other Android devices (except for HTC and Motorola). For a more detailed explanation behind this exploit, check out Dallas' article for a full breakdown.
Point your phone's browser to the TowelRoot website to get started. From there, simply click on the lambda symbol to download the app—just make sure you have installation from "Unknown sources" enabled, which you can do through Menu -> Settings -> Security.
Once downloaded, install the app like any other.
Open the app, and hit make it ra1n to start the rooting process.
After that...well, you're done. Really. That's it.
TowelRoot exploits all Android kernels built before June 3rd, 2014, and during the exploit, the app injects root. This is a new method of rooting, and one that hasn't yet been patched. So yes, you really did just root your Note 2 with a click-and-reboot.
Once you've got the app installed, run it and install the binaries it needs, and you'll be all set.
That was easy, and that's the beauty of Android.