Trick I employ to get my Bluetooth headset to work
My PRIV is from VZW and is still running Lollipop.
My Bluetooth headset is a Plantronics Voyager Legend (best in the world in my book and I've tried way too many and done far too much research), which I've used for years with my other BB phones with no issue.
With the PRIV, however, at first it seemed it would connect and disconnect at will on starting a call. Then it seemed to maintain pretty consistent faulty behavior -- it would work if it was already connected before the call, but if I connected it after I was on a call the nice female computer voice in the headset would say it was connected and the phone's icon would indicate it was connected, but the speech in both directions would be coming through the handset.
The trick I discovered as a workaround is to tap to tell the phone to use the Speakerphone and then tap to tell it to use Bluetooth. That seems to do the trick every time.
(Telling it to use the Handset and then Bluetooth doesn't seem to work, or at least not as consistently.)