Weakly supervised semantic segmentation (WSSS) aims at learning a semantic segmentation model with only image-level tags. Despite intensive research on deep learning approaches over a decade, there is still a significant performance gap between WSSS and full semantic segmentation. Most current WSSS methods always focus on a limited single image (pixel-wise) information while ignoring the valuable interimage (semantic-wise) information. From this perspective, a novel end-to-end WSSS framework called DSCNet is developed along with two innovations: i) pixel-wise group contrast and semantic-wise graph contrast are proposed and introduced into the WSSS framework; ii) a novel dual-stream contrastive learning mechanism is designed to jointly handle pixel-wise and semantic-wise context information for better WSSS performance. Specifically, the pixel-wise group contrast learning and semantic-wise graph contrast learning tasks form a more comprehensive solution. Extensive experiments on PASCAL VOC and MS COCO benchmarks verify the superiority of DSCNet over SOTA approaches and baseline models.