Uh, that's the point. Otherwise why bother? Looking back at the YouTube Terms of Service (and others) anyone, can do anything with your video. You give them that permission when you post your footage with the website. If you don't "mess it up" then you have no protection because putting the watermark in a corner is ineffective.
Why would you watermark your still images in the center of the frame but not your video? What makes video different?