We consider a multiterminal source coding problem in which a random source signal is estimated from encoded versions of multiple noisy observations. Each encoded version, however, is compressed so as to minimize a local distortion measure, defined only with respect to the distribution of the corresponding noisy observation. The original source is then estimated from these compressed noisy observations. We denote the minimal distortion under this coding scheme as the compress-and-estimate distortion-rate function (CE-DRF). We derive a single-letter expression for the CE-DRF in the case of an i.i.d source. We evaluate this expression for the case of a Gaussian source observed through multiple parallel AWGN channels and quadratic distortion and in the case of a non-uniform binary i.i.d source observed through multiple binary symmetric channels under Hamming distortion. For the case of a Gaussian source, we compare the performance for centralized encoding versus that of distributed encoding. In the centralized encoding scenario, when the code rates are sufficiently small, there is no loss of performance compared to the indirect source coding distortion-rate function, whereas distributed encoding achieves distortion strictly larger then the optimal multiterminal source coding scheme. For the case of a binary source, we show that even with a single observation, the CE-DRF is strictly larger than that of indirect source coding.