Despite echo chambers in social media have been under considerable scrutiny, general models for their detection and analysis are missing. In this work, we aim to fill this gap by proposing a probabilistic generative model that explains social media footprints -- i.e., social network structure and propagations of information -- through a set of latent communities, characterized by a degree of echo-chamber behavior and by an opinion polarity. Specifically, echo chambers are modeled as communities that are permeable to pieces of information with similar ideological polarity, and impermeable to information of opposed leaning: this allows discriminating echo chambers from communities that lack a clear ideological alignment. To learn the model parameters we propose a scalable, stochastic adaptation of the Generalized Expectation Maximization algorithm, that optimizes the joint likelihood of observing social connections and information propagation. Experiments on synthetic data show that our algorithm is able to correctly reconstruct ground-truth latent communities with their degree of echo-chamber behavior and opinion polarity. Experiments on real-world data about polarized social and political debates, such as the Brexit referendum or the COVID-19 vaccine campaign, confirm the effectiveness of our proposal in detecting echo chambers. Finally, we show how our model can improve accuracy in auxiliary predictive tasks, such as stance detection and prediction of future propagations.