For distributed estimation in wireless sensor networks (WSNs), the estimation performance suffers from the presence of observation and communication noise, as well as the limited communication bandwidth between the sensor nodes (SNs) and the fusion center (FC). In this work, we propose an efficient distributed estimation scheme for WSNs, improving both the bandwidth and energy efficiency. In the proposed scheme, a multi-bit uniform quantizer with Gray codeword mapping is applied at each node. For each node, however, only one bit is chosen for transmission to the FC, in order to save the energy consumption and bandwidth utilization. By carefully pre-arranging the transmitted bits among all nodes, the information bits received at the FC can be successfully aggregated and fused to obtain the estimated signal and to overcome the impairment caused by the communication channel. The transmitted bits arrangement is formulated as a constrained optimization problem for resource allocation, with the object of minimizing the estimation mean square error (MSE) at the FC.