Deep learning's preponderance across scientific domains has reshaped high-stakes decision-making, making it essential to follow rigorous operational frameworks that include both Right-to-Privacy (RTP) and Right-to-Explanation (RTE). This paper examines the complexities of combining these two requirements. For RTP, we focus on `Differential privacy` (DP), which is considered the current gold standard for privacy-preserving machine learning due to its strong quantitative guarantee of privacy. For RTE, we focus on post-hoc explainers: they are the go-to option for model auditing as they operate independently of model training. We formally investigate DP models and various commonly-used post-hoc explainers: how to evaluate these explainers subject to RTP, and analyze the intrinsic interactions between DP models and these explainers. Furthermore, our work throws light on how RTP and RTE can be effectively combined in high-stakes applications. Our study concludes by outlining an industrial software pipeline, with the example of a wildly used use-case, that respects both RTP and RTE requirements.