You can actually see the perspective shift between the two images the 3DS displays simultaneously if you close one eye and turn the screen back and forth like a lensatic image. Do this with various photos taken with the 3DS (of varying distances from the two cameras) and you'll understand how the effect works. It's the same effect you get if you alternate between looking at anything from just one eye and then the other back and forth. As soon as you're looking through both eyes again, depth perception returns.
Unless you're only capable of seeing either the left or right eye image for a given frame, or one eye is significantly stronger than the other, the effect will be the perception of depth. So if the 3D effect doesn't work, or is weak, it's not the display; it's the viewer.
Technically speaking, the human eye only sees flat images projected onto the back of the retina. It's only when the brain combines both left and right eye images simultaneously that a person is able to perceive depth.
If you're getting this effect when staring at a flat, non stereoscopic image, you're not perceiving 3D; you're seeing double.
