Adaptive immunity’s success relies on the extraordinary diversity of protein receptors on B and T cell membranes. Despite this diversity, the existence of public receptors shared by many individuals gives hope for developing population-wide vaccines and therapeutics. Using probabilistic modeling, we show many of these public receptors are shared by chance in healthy individuals. This predictable overlap is driven not only by biases in the random generation process of receptors, as previously reported, but also by their common functional selection. However, the model underestimates sharing between repertoires of individuals infected with SARS-CoV-2, suggesting strong specific antigen-driven convergent selection. We exploit this discrepancy to identify COVID-associated receptors, which we validate against datasets of receptors with known viral specificity. We study their properties in terms of sequence features and network organization, and use them to design an accurate diagnostic tool for predicting SARS-CoV-2 status from repertoire data.