Conventional frame-based cameras often struggle with stereo depth estimation in rapidly changing scenes. In contrast, bio-inspired spike cameras emit asynchronous events at microsecond-level resolution, providing an alternative sensing modality. However, existing methods lack specialized stereo algorithms and benchmarks tailored to the spike data. To address this gap, we propose SpikeStereoNet, a brain-inspired framework and the first to estimate stereo depth directly from raw spike streams. The model fuses raw spike streams from two viewpoints and iteratively refines depth estimation through a recurrent spiking neural network (RSNN) update module. To benchmark our approach, we introduce a large-scale synthetic spike stream dataset and a real-world stereo spike dataset with dense depth annotations. SpikeStereoNet outperforms existing methods on both datasets by leveraging spike streams' ability to capture subtle edges and intensity shifts in challenging regions such as textureless surfaces and extreme lighting conditions. Furthermore, our framework exhibits strong data efficiency, maintaining high accuracy even with substantially reduced training data. The source code and datasets will be publicly available.