Join Date:Joined: Sep 2010
You should technically be able to install either as long as you don't flash a T-Mobile modem. I have had an AT&T Note 2 and still have an AT&T S4 (currently using a T-Mobile Note 3 though) all on T-Mobile. I haven't tried this on an AT&T Note 3, but on my AT&T Note 2 and S4, I was able to flash T-Mobile ROMs to gain Wifi Calling (calls, texts, and MMS work over Wifi, even overseas), which is fantastic because I have like no T-Mobile service in my house, but it's perfect when I walk outside. Additionally, you gain the "Network Mode" menu which AT&T removes from all of its devices.
Currently, I'm running Vision X N3 on my Note 3. Worst case, you get bootloops or freezing if it doesn't work on your AT&T model, but that's it. Just don't flash any T-Mobile bootloaders or modems (99% of ROMs for any version of the phone include neither).