I have a dual monitor setup with Linux where one display is connected via DisplayPort and the other via DVI, both from an Nvidia graphics card. When the screen is locked both displays go black, the DVI monitor says it’s entering power save mode and the DisplayPort monitor says “No input” at which point both monitors turn back on displaying the unlock screen.

Playing with xset dpms force standby|suspend|off gave a variety of effects, sometimes the DisplayPort stayed off but the DVI turned back on, sometimes the DisplayPort went black but didn’t turn off etc.

Ultimately though the solution was to disable the “DP Auto Switch” and “HDMI Auto Switch” setting on the DisplayPort monitor. I imagine on other monitors this is equivalent to auto detecting the input source. Now the default Linux settings do the right thing and turn both monitors off after a while, they stay off until I move the mouse and then both turn back on perfectly.