Weakly supervised referring expression comprehension (WREC) and segmentation (WRES) aim to learn object grounding based on a given expression using weak supervision signals like image-text pairs. While these tasks have traditionally been modeled separately, we argue that they can benefit from joint learning in a multi-task framework. To this end, we propose WeakMCN, a novel multi-task collaborative network that effectively combines WREC and WRES with a dual-branch architecture. Specifically, the WREC branch is formulated as anchor-based contrastive learning, which also acts as a teacher to supervise the WRES branch. In WeakMCN, we propose two innovative designs to facilitate multi-task collaboration, namely Dynamic Visual Feature Enhancement (DVFE) and Collaborative Consistency Module (CCM). DVFE dynamically combines various pre-trained visual knowledge to meet different task requirements, while CCM promotes cross-task consistency from the perspective of optimization. Extensive experimental results on three popular REC and RES benchmarks, i.e., RefCOCO, RefCOCO+, and RefCOCOg, consistently demonstrate performance gains of WeakMCN over state-of-the-art single-task alternatives, e.g., up to 3.91% and 13.11% on RefCOCO for WREC and WRES tasks, respectively. Furthermore, experiments also validate the strong generalization ability of WeakMCN in both semi-supervised REC and RES settings against existing methods, e.g., +8.94% for semi-REC and +7.71% for semi-RES on 1% RefCOCO.