I think most people aren't consciously aware of their eye's shallow DOF, but only in so much as they don't have the structural framework to consider it. We have the vocabulary and reference points to discuss it. This does not mean, however, that they don't see it. My wife certainly sees it in photos, sometimes says things like "I want a camera that makes my pictures look better" and then shows me images with shallow DOF emphasising the subject and blurring an otherwise disturbing background; and she realizes her own vision is like that. Though she has no language for it, she easily sees the difference between an 8x10 print from a 4x5 and one from a d-slr. These are not things many photographers believe the naive viewer of their works can see- I hold that most viewers do, they just have no ability to provide a reference point and language for what they see.
Also, I disagree with the notion that shallow DOF is a tool the average viewer doesn't see often. Most people watch a couple of movies a week, and they see quite a bit of imagery with shallow DOF, it is a common cinematic tool. Cinema is photography at 24 pictures a second with soundtrack... I don't think that enough photographers consider how cinema shapes their viewers' perception of imagery. Perhaps the generally high production value of cinematic imagery causes a false perception of value even with inappropriate use of shallow DOF.