I admit, I like to take videos of my dogs sleeping with my Motorola Droid. The sound it makes when you start recording video always wakes them up though. The same thing goes for the shutter sound for the still camera.

In order to disable the annoying sound, you have to exit out of the Camera app. Then from the Home screen, you can use the side buttons to control the ringer volume. Set the ringer to silent. Now when you open the Camera app again to take a picture or video, no sound will be made.