I do not mean "see photon shot noise" in representations of the world made by imaging devices.
I do mean "see photon shot noise" utilizing one's very own eyes in the physical world around us.
Great question.
I think the answer is that our visual system is designed to reject photon shot noise in much the same way that it rejects object color changes caused by tinted lighting -- or interpolates over the eye's blind spot. In other words, we certainly see it, but process to remove it from our perception.
Actually, to me this is one of the biggest problems with the digital imaging community: photography (and vision) isn't really about recording light, but about constructing a model of the scene. There is very little "survival value" to perception of statistical variations in lighting, so we generally don't perceive it. However, distinguishing between a tasty kind of fruit and a posionous one with a slightly different hue -- even when the lighting has changed so that the poisonous one has the same color as the tasty kind has in normal daylight -- well, that has obvious survival value. In fact, recognizing that an object has a continuous surface despite the fact that the photons that reveal it to us are intermittent (due to photon shot noise) gives survival value to not perceiving photo shot noise.
In extremely dim lighting the flaws in our perceptive processing do occasionally reveal artifacts that I suspect are photon shot noise overpowering our ability to make a good model. For example, I've often noticed that my perception of surfaces as continuous is often disrupted in very dim light, e.g., a solid wall will sometimes appear to be several disjoint pieces in very dim light.
