In this work we design an end-to-end deep learning architecture for predicting, on Chest X-rays images (CXR), a multi-regional score conveying the degree of lung compromise in COVID-19 patients. Such semi-quantitative scoring system, namely Brixia score, is applied in serial monitoring of such patients, showing significant prognostic value, in one of the hospitals that experienced one of the highest pandemic peaks in Italy. To solve such a challenging visual task, we adopt a weakly supervised learning strategy structured to handle different tasks (segmentation, spatial alignment, and score estimation) trained with a "from-the-part-to-the-whole" procedure involving different datasets. In particular, we exploit a clinical dataset of almost 5,000 CXR annotated images collected in the same hospital. Our BS-Net demonstrates self-attentive behavior and a high degree of accuracy in all processing stages. Through inter-rater agreement tests and a gold standard comparison, we show that our solution outperforms single human annotators in rating accuracy and consistency, thus supporting the possibility of using this tool in contexts of computer-assisted monitoring. Highly resolved (super-pixel level) explainability maps are also generated, with an original technique, to visually help the understanding of the network activity on the lung areas. We also consider other scores proposed in literature and provide a comparison with a recently proposed non-specific approach. We eventually test the performance robustness of our model on an assorted public COVID-19 dataset, for which we also provide Brixia score annotations, observing good direct generalization and fine-tuning capabilities that highlight the portability of BS-Net in other clinical settings. The CXR dataset along with the source code and the trained model are publicly released for research purposes.