In this paper, we investigate the performance of distributed estimation schemes in a wireless sensor network in the presence of an eavesdropper. The sensors transmit observations to the fusion center (FC), which at the same time are overheard by the eavesdropper. Both the FC and the eavesdropper reconstruct a minimum mean-squared error estimate of the physical quantity observed. We address the problem of transmit power allocation for system performance optimization subject to a total average power constraint on the sensor(s), and a security/secrecy constraint on the eavesdropper. We mainly focus on two scenarios: a single sensor with multiple transmit antennas and multiple sensors with each sensor having a single transmit antenna. For each scenario, given perfect channel state information (CSI) of the FC and full or partial CSI of the eavesdropper, we derive the transmission policies for short-term and long-term cases. For the long-term power allocation case, when the sensor is equipped with multiple antennas, we can achieve zero information leakage in the full CSI case, and dramatically enhance the system performance by deploying the artificial noise technique for the partial CSI case. Asymptotic expressions are derived for the long-term distortion at the FC as the number of sensors or the number of antennas becomes large. In addition, we also consider multiple-sensor multiple-antenna scenario, and simulations show that given the same total number of transmitting antennas the multiple-antenna sensor network is superior to the performance of the multiple-sensor single-antenna network.