Event Stream Super-Resolution (ESR) aims to address the challenge of insufficient spatial resolution in event streams, which holds great significance for the application of event cameras in complex scenarios. Previous works for ESR often process positive and negative events in a mixed paradigm. This paradigm limits their ability to effectively model the unique characteristics of each event and mutually refine each other by considering their correlations. In this paper, we propose a bilateral event mining and complementary network (BMCNet) to fully leverage the potential of each event and capture the shared information to complement each other simultaneously. Specifically, we resort to a two-stream network to accomplish comprehensive mining of each type of events individually. To facilitate the exchange of information between two streams, we propose a bilateral information exchange (BIE) module. This module is layer-wisely embedded between two streams, enabling the effective propagation of hierarchical global information while alleviating the impact of invalid information brought by inherent characteristics of events. The experimental results demonstrate that our approach outperforms the previous state-of-the-art methods in ESR, achieving performance improvements of over 11% on both real and synthetic datasets. Moreover, our method significantly enhances the performance of event-based downstream tasks such as object recognition and video reconstruction.