In this paper, a quasi-decentralized estimation framework for Wireless Sensor Networks (WSNs) is presented, where a fusion center (FC) collects measurements from nearby sensor nodes (SNs) to track a time-correlated random process. The SNs jointly adapt their data reporting rule based on the estimation quality feedback broadcast by the FC, with the overall objective to minimize the Mean Squared estimation Error at the FC, with a constraint on the sensing/transmission cost of each SN. It is proved that the special structure of the problem allows a sequential decomposition via Dynamic Programming, and that the optimal policy has a threshold structure with respect to the measurement accuracy of each SN. Moreover, owing to the homogeneity of the WSN setting considered, the optimality of symmetric policies is proved, thus enabling a reduction of the action space and of the overall optimization complexity. The numerical results demonstrate that, when the random process exhibits strong time-correlation and the SNs are energy constrained, a non-trivial performance gain is attained by performing adaptation to the estimation quality feedback broadcast by the FC.