This paper investigates the distributed Kalman filter (DKF) for linear systems, with specific attention on measurement fusion, which is a typical way of information sharing and is vital for enhancing stability and improving estimation accuracy. We show that it is the mismatch between the fused measurement and the fused covariance that leads to performance degradation or inconsistency in previous consensus-based DKF algorithms. To address this issue, we introduce two fully distributed approaches for calculating the exact covariance of the fused measurements, building upon which the modified DKF algorithms are proposed. Moreover, the performance analysis of the modified algorithms is also provided under rather mild conditions, including the steady-state value of the estimation error covariance. We also show that due to the guaranteed consistency in the modified DKF algorithms, the steady-state estimation accuracy is significantly improved compared to classical DKF algorithms. Numerical experiments are carried out to validate the theoretical analysis and show the advantages of the proposed methods.