Many data producers seek to provide users access to confidential data without unduly compromising data subjects' privacy and confidentiality. When intense redaction is needed to do so, one general strategy is to require users to do analyses without seeing the confidential data, for example, by releasing fully synthetic data or by allowing users to query remote systems for disclosure-protected outputs of statistical models. With fully synthetic data or redacted outputs, the analyst never really knows how much to trust the resulting findings. In particular, if the user did the same analysis on the confidential data, would regression coefficients of interest be statistically significant or not? We present algorithms for assessing this question that satisfy differential privacy. We describe conditions under which the algorithms should give accurate answers about statistical significance. We illustrate the properties of the methods using artificial and genuine data.