Lately, differential privacy (DP) has been introduced in cooperative multiagent reinforcement learning (CMARL) to safeguard the agents' privacy against adversarial inference during knowledge sharing. Nevertheless, we argue that the noise introduced by DP mechanisms may inadvertently give rise to a novel poisoning threat, specifically in the context of private knowledge sharing during CMARL, which remains unexplored in the literature. To address this shortcoming, we present an adaptive, privacy-exploiting, and evasion-resilient localized poisoning attack (PeLPA) that capitalizes on the inherent DP-noise to circumvent anomaly detection systems and hinder the optimal convergence of the CMARL model. We rigorously evaluate our proposed PeLPA attack in diverse environments, encompassing both non-adversarial and multiple-adversarial contexts. Our findings reveal that, in a medium-scale environment, the PeLPA attack with attacker ratios of 20% and 40% can lead to an increase in average steps to goal by 50.69% and 64.41%, respectively. Furthermore, under similar conditions, PeLPA can result in a 1.4x and 1.6x computational time increase in optimal reward attainment and a 1.18x and 1.38x slower convergence for attacker ratios of 20% and 40%, respectively.