Local depth variation is a distinctive property of natural scenes, but its effects on perception have only recently begun to be investigated. Depth variation in natural scenes is due to depth edges between objects and surface nonuniformities within objects. Here, we demonstrate how natural depth variation impacts performance in two fundamental tasks related to stereopsis: half-occlusion detection and disparity detection. We report the results of a computational study that uses a large database of natural stereo-images and coregistered laser-based distance measurements. First, we develop a procedure for precisely sampling stereo-image patches from the stereo-images and then quantify the local depth variation in each patch by its disparity contrast. Next, we show that increased disparity contrast degrades half-occlusion detection and disparity detection performance and changes the size and shape of the spatial integration areas ("receptive fields") that optimize performance. Then, we show that a simple image-computable binocular statistic predicts disparity contrast in natural scenes. Finally, we report the most likely spatial patterns of disparity variation and disparity discontinuities (half-occlusions) in natural scenes. Our findings motivate computational and psychophysical investigations of the mechanisms that underlie stereo processing tasks in local regions of natural scenes.