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In this paper, we address the problem of decentralized parameter estimation via Wireless Sensor Networks (WSNs). We consider two different encoding strategies, namely, Quantize-and-Estimate (Q&E) and Compress-and-Estimate (C&E) and assume that sensor observations are conveyed to the Fusion Center (FC) over a number of orthogonal Gaussian or Rayleigh-fading channels. We constrain both power and bandwidth to be constant irrespectively of the network size and find approximate closed-form expressions of the optimal number of sensors for a number of cases of interest. Besides, we derive the optimal encoding rate for the Q&E scheme when, in the absence of Transmit Channel State Information (CSIT), sensors must encode their observations at a common and constant rate. For the (successive) C&E strategy, we also determine the encoding order that minimizes the resulting distortion in the FC estimates. We complement the analysis by deriving an expression of the asymptotic distortion when the number of sensors grows without bound, and the rate at which distortion decreases in the high-SNR regime. Finally, we close the paper by presenting some computer simulation results.