A fully distributed version of sequential change detection is studied, where a bunch of distributed sensors tries to collaboratively detect the occurrence of an abrupt event as quickly as possible in the absence of a fusion center. Every link connecting two nodes is subject to a finite bandwidth constraint. A novel stopping rule is proposed where each sensor computes a modified version of the well-known cumulative sum (CUSUM) statistic, exchanges a quantized version of this modified CUSUM statistic with its neighbors via links with a finite bandwidth, and then decides its view of whether the event has occurred. Using the trade-off between the worst-case expected detection delay (EDD) and average run length (ARL) of a false alarm as the performance metric, the proposed algorithm is shown, via extensive simulations, to outperform the current state-of-the-art based on average consensus. Moreover, the proposed stopping rule can adjust the quantization levels to comply with the bandwidth constraint and only exchanges information occasionally, which is more energy- efficient and much more spectrum-efficient than the one based on average consensus requiring infinite bandwidth and constantly exchanging information.