Modern real-time Structural Health Monitoring systems can generate a considerable amount of information that must be processed and evaluated for detecting early anomalies and generating prompt warnings and alarms about the civil infrastructure conditions. The current cloud-based solutions cannot scale if the raw data has to be collected from thousands of buildings. This paper presents a full-stack deployment of an efficient and scalable anomaly detection pipeline for SHM systems which does not require sending raw data to the cloud but relies on edge computation. First, we benchmark three algorithmic approaches of anomaly detection, i.e., Principal Component Analysis (PCA), Fully-Connected AutoEncoder (FC-AE), and Convolutional AutoEncoder (C-AE). Then, we deploy them on an edge-sensor, the STM32L4, with limited computing capabilities. Our approach decreases network traffic by $\approx8\cdot10^5\times$ , from 780KB/hour to less than 10 Bytes/hour for a single installation and minimize network and cloud resource utilization, enabling the scaling of the monitoring infrastructure. A real-life case study, a highway bridge in Italy, demonstrates that combining near-sensor computation of anomaly detection algorithms, smart pre-processing, and low-power wide-area network protocols (LPWAN) we can greatly reduce data communication and cloud computing costs, while anomaly detection accuracy is not adversely affected.