Since the advent of deep learning (DL), the field has witnessed a continuous stream of innovations. However, the translation of these advancements into practical applications has not kept pace, particularly in safety-critical domains where artificial intelligence (AI) must meet stringent regulatory and ethical standards. This is underscored by the ongoing research in eXplainable AI (XAI) and privacy-preserving machine learning (PPML), which seek to address some limitations associated with these opaque and data-intensive models. Despite brisk research activity in both fields, little attention has been paid to their interaction. This work is the first to thoroughly investigate the effects of privacy-preserving techniques on explanations generated by common XAI methods for DL models. A detailed experimental analysis is conducted to quantify the impact of private training on the explanations provided by DL models, applied to six image datasets and five time series datasets across various domains. The analysis comprises three privacy techniques, nine XAI methods, and seven model architectures. The findings suggest non-negligible changes in explanations through the implementation of privacy measures. Apart from reporting individual effects of PPML on XAI, the paper gives clear recommendations for the choice of techniques in real applications. By unveiling the interdependencies of these pivotal technologies, this research marks an initial step toward resolving the challenges that hinder the deployment of AI in safety-critical settings.