The Chromecast does not require external power; it'll run off the power supplied via the HDMI port. I'm pretty sure the Amazon FireTV Stick will, too. I don't own a Roku Stick, so I can't comment.
They prefer having external power because many HDMI ports don't supply enough consistently, but don't require it.
HDMI provides about 50mA minimum. USB can provide up to 500mA or 900mA depending on USB 2 or 3.
I am not 100% sure what the power requirements for a Chromecast are, I'm 100% sure it is more then 50mA.
If you are plugging in your Chromecast to your TV without the USB and it is working it's because you have a MHL-HDMI that provides up to 900mA of power.
They prefer having external power because many HDMI ports don't supply enough consistently, but don't require it.