We consider the multiterminal remote source coding problem of estimating a Gaussian signal from a bit-restricted representation of distributed linear measurements corrupted by additive white Gaussian noise. For this problem, we study the performance of the multiterminal compress-and-estimate (CE) coding scheme in which multiple remote encoders compress their measurements so as to minimize a local distortion measure which depends solely on the distribution of these measurements. In reconstruction, the decoder estimates the signal from the lossy-compressed measurements having full knowledge of the statistics of the source signal and the noisy measurements. The CE coding scheme is motivated by the scenario in which source encoders, due to their limited capabilities, operate according to a pre-determined compression strategy and cannot adapt to the sensing environment while the fusion center has full knowledge and computational capabilities. We focus, in particular, on two scenarios: the centralized observation model in which measurements are collected at a single remote encoder and the distributed observation model where measurements are provided to multiple remote sensors. In both scenarios, we investigate the performance attainable through the CE coding scheme in which the measurements are compressed according to a quadratic distortion measure and compare it to the performance of the coding scheme having full system knowledge.