In this paper, we analyze two coding strategies, namely, quantize-and-estimate (Q&E) and compress-and-estimate (C&E) which are suitable for decentralized parameter estimation with wireless sensor networks (WSNs). In the Q&E approach, the sensors encode their observations irrespectively of any statistical information that could be made available e.g. by the fusion center (FC). On the contrary, C&E effectively incorporates side information into the successive encoding process, which turns out to be very useful in the case of correlated observations that we address. In a context of Gaussian channels, we impose a total bandwidth (and power) constraint, this allowing us to assess the corresponding performance vs. number of sensors trade-offs in a more realistic manner. Performance is assessed by means of computer simulations and, also, by deriving some closed-form expressions of the estimation accuracy achievable for an asymptotically high number of sensors.