In this paper, we investigate a class of distortion outage minimization problems for a wireless sensor network in the presence of an eavesdropper. The observation signals transmitted from the sensors to the fusion center (FC) are overheard by the eavesdropper. Both the FC and the eavesdropper reconstruct minimum mean squared error estimates of the physical quantity observed. We address the problem of transmit power allocation to minimize the distortion outage at the FC, subject to an expected total transmit power constraint across the sensor(s) and a secrecy outage constraint at the eavesdropper. Applying a rigorous probabilistic power allocation technique, we derive power policies for the full channel state information (CSI) case. Suboptimal power control policies are studied for the partial CSI case in order to reduce the high computational cost associated with large numbers of sensors or receive antennas. Numerical results show that significantly improved performance can be achieved by adding multiple receive antennas at the FC. In the case of multiple transmit antennas, the distortion outage at the FC can be dramatically reduced and in some cases completely eliminated, by transmitting the observations on the null space of the eavesdropper's channel or deploying an artificial noise technique, in the full CSI and partial CSI cases, respectively.