Considerable effort has been devoted to developing a comprehensive understanding of CRISPR nuclease specificity. In silico predictions and multiple genome-wide cellular and biochemical approaches have revealed a basic understanding of the Cas9 specificity profile. However, none of these approaches has delivered a model that allows accurate prediction of a CRISPR nuclease's ability to cleave a site based entirely on the sequence of the guide RNA (gRNA) and the target. We describe a library-based biochemical assay that directly reports the cleavage efficiency of a particular Cas9-guide complex by measuring both uncleaved and cleaved target molecules over a wide range of mismatched library members. We applied our assay using libraries of targets to evaluate the specificity of Staphylococcus aureus Cas9 under a variety of experimental conditions. Surprisingly, our data show an unexpectedly high variation in the random gRNA:target DNA mismatch tolerance when cleaving with different gRNAs, indicating guide-intrinsic mismatch permissiveness and challenging the assumption of universal specificity models. We use data generated by our assay to create the first off-target, guide-specific cleavage models. The barcoded libraries of targets approach is rapid, highly modular, and capable of generating protein- and guide-specific models, as well as illuminating the biophysics of Cas9 binding versus cutting. These models may be useful in identifying potential off-targets, and the gRNA-intrinsic nature of mismatch tolerance argues for coupling these specificity models with orthogonal methods for a more complete assessment of gRNA specificity.