Surveillance of SARS-CoV-2 through reported positive RT-PCR tests is biased due to non-random testing. Prevalence estimation in population-based samples corrects for this bias. Within this context, the pooled testing design offers many advantages, but several challenges remain with regards to the analysis of such data. We developed a Bayesian model aimed at estimating the prevalence of infection from repeated pooled testing data while (i) correcting for test sensitivity; (ii) propagating the uncertainty in test sensitivity; and (iii) including correlation over time and space. We validated the model in simulated scenarios, showing that the model is reliable when the sample size is at least 500, the pool size below 20, and the true prevalence below 5%. We applied the model to 1.49 million pooled tests collected in Switzerland in 2021-2022 in schools, care centres, and workplaces. We identified similar dynamics in all three settings, with prevalence peaking at 4-5% during winter 2022. We also identified differences across regions. Prevalence estimates in schools were correlated with reported cases, hospitalizations, and deaths (coefficient 0.84 to 0.90). We conclude that in many practical situations, the pooled test design is a reliable and affordable alternative for the surveillance of SARS-CoV-2 and other viruses.