This paper considers a distributed filtering problem over a multi-sensor network in which the correlation of local estimation errors is unknown. Recently, this problem was studied by G. Battistelli  by developing a data fusion rule to calculate the weighted Kullback-Leibler average of local estimates with consensus algorithms for distributed averaging, where the weighted Kullback-Leibler average is defined as an averaged probability density function to minimize the sum of weighted Kullback-Leibler divergences from the original probability density functions. In this paper, we extends those earlier results by relaxing the prior assumption that all sensors share the same degree of confidence. Furthermore, a novel consensus-based distributed weighting coefficients selection scheme is developed to improve the fusion accuracy, where the weight associated with each sensor is adjusted based on the local estimation error covariance and the ones received from neighboring sensors, so that larger weight values will be assigned to a sensor with higher degree of confidence. Finally, a Monte-Carlo simulation with a 2D tracking system validates the effectiveness of the proposed distributed filtering algorithm.