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Streaming data collection is essential to real-time data analytics in various IoTs and mobile device-based systems, which, however, may expose end users' privacy. Local differential privacy (LDP) is a promising solution to privacy-preserving data collection and analysis. However, existing few LDP studies over streams are either applicable to finite streams only or suffering from insufficient protection. This paper investigates this problem by proposing LDP-IDS, a novel $w$-event LDP paradigm to provide practical privacy guarantee for infinite streams at users end, and adapting the popular budget division framework in centralized differential privacy (CDP). By constructing a unified error analysi for LDP, we first develop two adatpive budget division-based LDP methods for LDP-IDS that can enhance data utility via leveraging the non-deterministic sparsity in streams. Beyond that, we further propose a novel population division framework that can not only avoid the high sensitivity of LDP noise to budget division but also require significantly less communication. Based on the framework, we also present two adaptive population division methods for LDP-IDS with theoretical analysis. We conduct extensive experiments on synthetic and real-world datasets to evaluate the effectiveness and efficiency pf our proposed frameworks and methods. Experimental results demonstrate that, despite the effectiveness of the adaptive budget division methods, the proposed population division framework and methods can further achieve much higher effectiveness and efficiency.