The paper demonstrates the effectiveness of the SNOWED dataset, specifically designed for identifying water bodies in Sentinel-2 images, in developing a remote sensing system based on deep neural networks. For this purpose, a system is implemented for monitoring the Po River, Italy's most important watercourse. By leveraging the SNOWED dataset, a simple U-Net neural model is trained to segment satellite images and distinguish, in general, water and land regions. After verifying its performance in segmenting the SNOWED validation set, the trained neural network is employed to measure the area of water regions along the Po River, a task that involves segmenting a large number of images that are quite different from those in SNOWED. It is clearly shown that SNOWED-based water area measurements describe the river status, in terms of flood or drought periods, with a surprisingly good accordance with water level measurements provided by 23 in situ gauge stations (official measurements managed by the Interregional Agency for the Po). Consequently, the sensing system is used to take measurements at 100 "virtual" gauge stations along the Po River, over the 10-year period (2015-2024) covered by the Sentinel-2 satellites of the Copernicus Programme. In this way, an overall space-time monitoring of the Po River is obtained, with a spatial resolution unattainable, in a cost-effective way, by local physical sensors. Altogether, the obtained results demonstrate not only the usefulness of the SNOWED dataset for deep learning-based satellite sensing, but also the ability of such sensing systems to effectively complement traditional in situ sensing stations, providing precious tools for environmental monitoring, especially of locations difficult to reach, and permitting the reconstruction of historical data related to floods and draughts. Although physical monitoring stations are designed for rapid monitoring and prevention of flood or other disasters, the developed tool for remote sensing of water bodies could help decision makers to define long-term policies to reduce specific risks in areas not covered by physical monitoring or to define medium- to long-term strategies such as dam construction or infrastructure design.