One of the challenges when it comes to switching gaming setups from the Windows world to Linux, is fully-featured hardware support.
The Xbox Wireless Headset + official dongle does a decent job with a lot of bang for the buck. However, It's not (yet) supported by XONE or any other driver. I can connect it via bluetooth, but then it just sounds dull - no surprise!
That's why I'm now looking for a new headset which is approved by the community. It must offer decent (surround-)sound in games.
For the most part these days, they do. But OP asked about wireless.
The problem with Bluetooth is not the operating system or drivers, but Bluetooth itself. The spec famously lacks provisions for good quality stereo output with good quality input at the same time. This is why many wireless headsets use a (non-Bluetooth) dongle.