This article studies a security issue in remote distributed consensus estimation where sensors transmit their measurements to remote estimators via a wireless communication network. The relative entropy is utilized as a stealthiness metric to detect whether the data transmitted through the wireless network are attacked. The performance degradation induced by an attacker that attempts to be stealthy or undetected is analyzed, and the corresponding false-data attack strategy is characterized, which can achieve the maximal integrated mean-square error (IMSE). Finally, the tradeoff between the performance degradation and attack stealthiness level is evaluated through an example.