We study the effects of decoding and processing costs in an energy harvesting two-way channel. We design the optimal offline power scheduling policies that maximize the sum throughput by a given deadline, subject to energy causality constraints, decoding causality constraints, and processing costs at both users. In this system, each user spends energy to transmit data to the other user, and also to decode data coming from the other user; that is, each user divides its harvested energy for transmission and reception. Further, each user incurs a processing cost per unit time as long as it communicates. The power needed for decoding the incoming data is modeled as an increasing convex function of the incoming data rate; and the power needed to be on, i.e., the processing cost, is modeled to be a constant per unit time. We solve this problem by first considering the cases with decoding costs only and processing costs only individually. In each case, we solve the single energy arrival scenario, and then use the solution's insights to provide an iterative algorithm that solves the multiple energy arrivals scenario. Then, we consider the general case with both decoding and processing costs in a single setting, and solve it for the most general scenario of multiple energy arrivals.