A wireless sensor network (WSN) engaged in a decentralized estimation problem is considered. The nonrandom unknown parameter lies in some small neighborhood of a nominal value and, exploiting this knowledge, a locally optimum estimator (LOE) is introduced. Under the LOE paradigm, the sensors of the network process their observations by means of a suitable nonlinearity (the score function), before delivering data to the fusion center that outputs the final estimate. Usually continuous-valued data cannot be reliably delivered from sensors to the fusion center, and some form of data compression is necessary. Accordingly, we design the scalar quantizers that must be used at the network's nodes in order to comply with the estimation problem at hand. Such a difficult multiterminal inference problem is shown to be asymptotically equivalent to the already solved problem of designing optimum quantizers for reconstruction (as opposed to inference) purposes.