I'm glad to see I'm not the only one experiencing this problem. I have a non-rooted Nexus 6 w/o a screen protector on Android 6.0. Seeing that it's not across the board, it might be related to a combination of apps or settings that may be causing the issue.
Similar symptoms experienced by others including:
1. Sometimes reverts to normal behaviour after a reboot, but soon reverts back. I'ts unclear under what conditions it reverts back eg. seems to be independent of the screensaver.
2. Can momentarily recover the screen if I press the power button repeatedly. If I'm lucky, I'm able to terminate a call when that happens. However, this is really dangerous to try to do while driving (of course the phone is in a car mount).
3. Outgoing calls are an issue - can't select the speakerphone or the kepad. Incoming calls can be answered (the screen is visible), but thereafter, when the call is answered, speakerphone and the keypad can't be selected because of the dark screen.
I tried removing apps with deep hooks into the OS including antivirus software Sophos Security and password keeper Lastpass, but no change.
I also tried removing phone type apps that may have conflicted with the phone app. which in my case is Fritz$Fon as I have a FritzBox router-modem, but again no change.