March 7, 2017 (this edit is no longer the answer):
No, Google Now ("Ok Google" command) is not broken when the phone is in sleep mode or when the phone is locked. Don't believe all of the forums. If you haven't asked a real software engineer (like myself) a question about phone configuration, don't trust the internet! Period! However, Android 6.0.1 has a slight bug. The "Always On" toggle switch is broken after an Android 6.0.1 upgrade. There are two ways to get to this from the Google Nexus 6P phone.
Settings > Language & Input > Google Voice Typing > "Ok Google" Detection > Always On (toggle)
Settings > Google > Search > Voice > "Ok Google" Detection > Always On (toggle)
If you don't reboot after the Android upgrade, this "Always On" option cannot be turned on. It's forever in the off position. To fix, reboot the phone, and then turn on the "Always On" toggle.
Wala! Now you should be able to take your phone out of sleep mode AND unlock your phone with your voice by saying "Ok Google ... Call {
contact
}
on speakerphone", assuming you've trained the phone. Google paid to patent this.
Note: I also have "Trusted Voice" turned off. I have NOT tested with this turned on yet. Will do that now.
March 13, 2017 (this edit is the new answer):
Google decided to remove this feature now. So now you get the "Actually I can't do that while the device is locked."
Settings > Google > Search > Voice > "Ok Google" Detection > Always On (toggle)
A Google manager must have read my post and told their team to disable this broken feature. So now the configuration (Always On) controlling this feature (that Google paid the patent for) is missing. So the Nexus 6P phone cannot be unlocked using this configuration any longer.
To unlock your phone with your voice:
Looks like Google decided to recently start using the Trusted Voice configuration to unlock the phone. This must be set up and turned on. Then you will get the functionality back.