Machine learning-based demodulation of multi-peak fiber Bragg grating (FBG) sensors has been extensively studied, demonstrating superior performance compared to conventional algorithms because it can neglect potential physical constraints. As the number of real-world monitoring points increases, the volume of fiber-optic sensing data grows exponentially. This necessitates aggregating data from various regions (e.g., via Wi-Fi), unlike traditional single-point monitoring, which challenges server storage capacity and communication efficiency. To address these issues, this paper proposes a federated learning (FL)-based framework for efficient wavelength demodulation of multi-peak FBGs in multipoint monitoring. Specifically, an arrayed waveguide grating (AWG) with multiplexing capability is employed at different monitoring points to convert spectral features into multi-channel transmission intensities, serving as training data for local models. Subsequently, the local model parameters, trained independently at each point, are uploaded to a central server to derive the optimal global model for demodulation across different monitoring points. The proposed demodulation framework is validated through stress demodulation experiments on multi-peak FBG sensors. Experimental results indicate strong multi-peak extraction performance with a demodulation error of ±0.64 pm. Additionally, the method demonstrates excellent applicability for generating high-precision global demodulation models through multi-node cooperative work under various scenarios.