We consider efficient study designs to estimate sensitivity and specificity of a candidate diagnostic or screening test. Our focus is the setting in which the candidate test is inexpensive to administer compared to evaluation of disease status, and the test results, available in a large cohort, can be used as a basis for sampling subjects for verification of disease status. We examine designs in which disease status is verified in a sample chosen so as to optimize estimation of either sensitivity or specificity. We then propose a sequential design in which the first step of sampling is conducted to efficiently estimate specificity. If the candidate test is determined to be of sufficient specificity, then step two of sampling is conducted to estimate sensitivity. We propose estimators based on this sequential sampling scheme, and show that the performance of these estimators is excellent. We develop sample size calculations for the sequential design, and show that this design, in most situations, compares favourably in terms of expected sample size to a fixed size design.