They made the unequivocal statement that it could be done while the computer was in sleep mode.
Nonsense.
You need the computer to be powered up because without the thing being alive and kicking, the network card won't be fully powered and there won't be much processing taking place taking the bits and bytes from the camera and pushing them down the network link.
In sleep mode, the only thing the computer handles are the low-level activities necessary to awake it from its slumber (i.e. mouse movements, keyboard button presses, specific network card packets).
Technically, "sleep mode" is defined by the industry standard ACPI S3 ... what happens is that the RAM is kept powered in a special low-power mode , but the CPU,cache, chipset and peripherals are powered down (context data from the CPU being saved to the RAM in the form of a "resume handler"). Google "ACPI S3" if you want more nitty gritty detail....
Probably what he meant was that you can wake a sleeping computer and by extension either send no signal to the display or tell it to remain dimmed to zero.
But to wake a computer, you have to be doing it from another device on the same local network (so called "magic packets" are strictly OSI layer 2) ... which means they would have hacked into your infrastructure pretty deeply and so them peering at your bedroom antics would be the least of your concerns.
P.S. Incase you are wondering, for ACPI S4 (or "hibernate") the CPU context gets saved to disk instead of RAM. The RAM then gets powered off too, making hibernate an extra low power standby mode.