Dual-pixel (DP) imaging sensors are getting more popularly adopted by modern cameras. A DP camera captures a pair of images in a single snapshot by splitting each pixel in half. Several previous studies show how to recover depth information by treating the DP pair as an approximate stereo pair. However, dual-pixel disparity occurs only in image regions with defocus blur which is unlike classic stereo disparity. Heavy defocus blur in DP pairs affects the performance of depth estimation approaches based on matching. Therefore, we treat the blur removal and the depth estimation as a joint problem. We investigate the formation of the DP pair, which links the blur and depth information, rather than blindly removing the blur effect. We propose a mathematical DP model that can improve depth estimation by the blur. This exploration motivated us to propose our previous work, an end-to-end DDDNet (DP-based Depth and Deblur Network), which jointly estimates depth and restores the image in a supervised fashion. However, collecting the ground-truth (GT) depth map for the DP pair is challenging and limits the depth estimation potential of the DP sensor. Therefore, we propose an extension of the DDDNet, called WDDNet (Weakly-supervised Depth and Deblur Network), which includes an efficient reblur solver that does not require GT depth maps for training. To achieve this, we convert all-in-focus images into supervisory signals for unsupervised depth estimation in our WDDNet. We jointly estimate an all-in-focus image and a disparity map, then use a Reblur and Fstack module to regularize the disparity estimation and image restoration. We conducted extensive experiments on synthetic and real data to demonstrate the competitive performance of our method when compared to state-of-the-art (SOTA) supervised approaches. Index Terms-Dual-pixel Sensor, Weakly-supervised, Depth Estimation, Deblur and Reblu.