How to measure the perceptual quality of depth information in stereoscopic natural images, especially images undergoing different types of symmetric and asymmetric distortions, is a fundamentally important issue that is not well understood. In this paper, we present two of our recent subjective studies on depth quality. The first one follows the absolute category rating (ACR) protocol that is widely used in general image quality assessment research. We find that traditional approaches such as ACR is problematic in this scenario because monocular cues and the spatial quality of images have strong impacts on the depth quality scores given by subjects, making it difficult to single out the actual contributions of stereoscopic cues in depth perception. To overcome this problem, we carry out the second subjective study where depth effect is synthesized at different depth levels before various types and levels of symmetric and asymmetric distortions are applied. Instead of following the traditional approach, we ask subjects to identify and label depth polarizations, and a Depth Perception Difficulty Index (DPDI) is developed based on the percentage of correct and incorrect subject judgements. We find this approach highly effective at quantifying depth perception induced by stereo cues and observe a number of interesting effects regarding image content dependency, distortion type dependency, and the impacts of symmetric versus asymmetric distortions. We believe that these are useful steps towards building comprehensive 3D quality-of-experience models for stereoscopic images.Index Terms-depth perception, stereoscopic image, 3D image, image quality assessment, quality-of-experience, asymmetric distortion