We introduce new binocular accuracy measures as alternatives to conventional marginal measures that can be used to evaluate screening tests in diagnostic studies involving paired organs (e.g. eyes and ears). Specifically, we consider screening studies based on a cross-sectional design, where both diagnosis and disease status are determined after study enrolment or sampling, yielding paired binocular binary data described via two models, namely, the extended common correlation model and the Gaussian copula probit model. The first relies on the assumption of exchangeability of fellow organs, while the second is more flexible. Binocular versions of sensitivity and specificity are defined, respectively, as the probability of at least one correct positive diagnosis in patients with one or both organs truly diseased and the probability of two correct negative diagnoses for patients with both organs truly un-diseased. Comparisons between the conventional marginal and binocular sensitivities and specificities are illustrated for both models using data from a diabetic retinopathy study. We show that our methodology provides a viable alternative to conventional ways of assessing diagnostic accuracy of screening tests for paired organs. The binocular versions of sensitivity and specificity reflect the way screening tests are conducted in practice, and they overcome the shortcomings of conventional measures. Copyright © 2017 John Wiley & Sons, Ltd.