Product retrieval systems have served as the main entry for customers to discover and purchase products online. With increasing concerns on the transparency and accountability of AI systems, studies on explainable information retrieval has received more and more attention in the research community. Interestingly, in the domain of e-commerce, despite the extensive studies on explainable product recommendation, the studies of explainable product search is still in an early stage. In this paper, we study how to construct effective explainable product search by comparing model-agnostic explanation paradigms with model-intrinsic paradigms and analyzing the important factors that determine the performance of product search explanations. We propose an explainable product search model with model-intrinsic interpretability and conduct crowdsourcing to compare it with the state-of-the-art explainable product search model with model-agnostic interpretability. We observe that both paradigms have their own advantages and the effectiveness of search explanations on different properties are affected by different factors. For example, explanation fidelity is more important for user's overall satisfaction on the system while explanation novelty may be more useful in attracting user purchases. These findings could have important implications for the future studies and design of explainable product search engines.