Hypothesis testing is a statistical inference framework for determining the true distribution among a set of possible distributions for a given dataset. Privacy restrictions may require the curator of the data or the respondents themselves to share data with the test only after applying a randomizing privacy mechanism. This work considers mutual information and maximal leakage as the privacy metrics for measuring leakage. In addition, motivated by the Chernoff-Stein lemma, the relative entropy between a pair of distributions of the output (generated by the privacy mechanism) is chosen as the utility metric. For these metrics, we find the optimal privacy-utility trade-off (PUT) and the corresponding optimal privacy mechanism in high privacy or utility regimes.