Energy consumption in wireless sensor networks is dominated by intra-network communication dedicated to coordination and information exchange between sensor nodes and the fusion center. The design of distributed estimation algorithms with reduced communication overheads is thus rather crucial. For amplify-and-forward sensor networks over flat fading channels, this paper proposes a new distributed best-linear-unbiased-estimation (BLUE) scheme by exploiting the statistical characterizations of the sensing noise variance and channel gains. The performance measure is the reciprocal of the mean square error averaged over the considered statistical distributions. We derive a closed-form lower bound for the adopted design metric. By means of this result, we further derive a closed-form universal sensor power amplification factor capable of maintaining a target estimation performance. The proposed scheme has the advantage that repeated power scheduling and message feedback are no longer needed in the parameter estimation phase and, hence, the in-network communication cost is further reduced. Some key features regarding the proposed method are discussed. Computer simulations are conducted to evidence our analytic study.