Speech enhancement (SE) improves communication in noisy environments, affecting areas such as automatic speech recognition, hearing aids, and telecommunications. With these domains typically being power-constrained and event-based while requiring low latency, neuromorphic algorithms in the form of spiking neural networks (SNNs) have great potential. Yet, current effective SNN solutions require a contextual sampling window imposing substantial latency, typically around 32ms, too long for many applications. Inspired by Dual-Path Spiking Neural Networks (DPSNNs) in classical neural networks, we develop a two-phase time-domain streaming SNN framework -- the Dual-Path Spiking Neural Network (DPSNN). In the DPSNN, the first phase uses Spiking Convolutional Neural Networks (SCNNs) to capture global contextual information, while the second phase uses Spiking Recurrent Neural Networks (SRNNs) to focus on frequency-related features. In addition, the regularizer suppresses activation to further enhance energy efficiency of our DPSNNs. Evaluating on the VCTK and Intel DNS Datasets, we demonstrate that our approach achieves the very low latency (approximately 5ms) required for applications like hearing aids, while demonstrating excellent signal-to-noise ratio (SNR), perceptual quality, and energy efficiency.