We investigate the weighted-sum distortion minimization problem in transmitting two correlated Gaussian sources over Gaussian channels using two energy harvesting nodes. To this end, we develop offline and online power control policies to optimize the transmit power of the two nodes. In the offline case, we cast the problem as a convex optimization and investigate the structure of the optimal solution. We also develop a generalized water-filling based power allocation algorithm to obtain the optimal solution efficiently. For the online case, we quantify the distortion of the system using a cost function and show that the expected cost equals the expected weighted-sum distortion. Based on Banach's fixed point theorem, we further propose a geometrically converging algorithm to find the minimum cost via simple iterations. Simulation results show that our online power control outperforms the greedy power control where each node uses all the available energy in each slot and performs close to that of the proposed offline power control. Moreover, the performance of our offline power control almost coincides with the performance limit of the system.