Sharing behavior on short-video platforms constitutes a complex ternary interaction among the user (sharer), the video (content), and the receiver. Traditional industrial solutions often decouple this into two independent tasks: video recommendation (predicting share probability) and receiver recommendation (predicting whom to share with), leading to suboptimal performance due to isolated modeling and inadequate information utilization. To address this, we propose UniShare, a novel unified framework for joint sharing prediction on both video and receiver recommendation. UniShare models the share probability through an enhanced representation learning module that incorporates pre-trained GNN and multi-modal embeddings, alongside explicit bilateral interest and relationship matching. A key innovation is our joint training paradigm, which leverages signals from both tasks to mutually enhance each other, mitigating data sparsity and improving bilateral satisfaction. We also introduce K-Share, a large-scale real-world dataset constructed from Kuaishou platform logs to support research in this domain. Extensive offline experiments demonstrate that UniShare significantly outperforms strong baselines on both tasks. Furthermore, online A/B testing on the Kuaishou platform confirms its effectiveness, achieving significant improvements in key metrics including the number of shares (+1.95%) and receiver reply rate (+0.482%).