which carrier does the Note 8 work on? TMobile or Verizon?

If you're on Oreo, it would automatically switch to T-Mobile firmware when you add the T-Mobile sim. All US Note 8's have the same hardware, so they can work on any carrier. I just picked up a Verizon Note 8 on Swappa and received it today, once I put my T-Mo sim in, it restarted and added T-Mo's apn, carrier splash screen on bootup, wi-fi calling feature, and bloat. I'm now flashing the unlocked firmware to get rid of all that though lol.
 
If you're on Oreo, it would automatically switch to T-Mobile firmware when you add the T-Mobile sim. All US Note 8's have the same hardware, so they can work on any carrier. I just picked up a Verizon Note 8 on Swappa and received it today, once I put my T-Mo sim in, it restarted and added T-Mo's apn, carrier splash screen on bootup, wi-fi calling feature, and bloat. I'm now flashing the unlocked firmware to get rid of all that though lol.
This is correct. I have an unlocked T-Mobile Note 8 and when I insert a AT&T sim, the phone reboots, shows the AT&T boot up logo as if it were an AT&T branded phone and everything works, including WiFi calling. Even all the AT&T bloat installs including DirecTV and the MyAT&T app.