While in-network caching is an essential feature of Information Centric Networks (ICN) for improved content dissemination and reducing the bandwidth consumption at the core of the network, it is prone to many privacy threats. For example, an adversary can passively breach the privacy of a consumer by simply analyzing the different retrieval times for the same content. This paper aims to address this problem of timing analysis attacks by developing privacy-enhancing caching strategies. The proposed caching strategies use two privacy metrics, namely mutual information from information theory and differential privacy, and formulates a privacy enhancing distributed optimization problem with the objective of optimizing the network cost incurred. We efficiently solve the optimization problem by considering it as a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -player, non-cooperative game. We show that Nash equilibrium exists for this game and compute it using an iterative best response algorithm. We compare and validate the performance of our approach on realistic network topologies by comparing it with the existing approaches in literature and the global optimal solutions.