In this work, we consider the amplify-and-forward (AF) two-way relay network where the two terminals TA, TB exchange their information through a relay node TR in a bi-directional manner and study the impact of the training-based channel estimation upon individual and sum-rate of the two users. We assume that in the multiple-access (MA) phase TA, TB initiate transmission by sending their training symbols to TR, and in the broadcasting (BC) phase TR initiates transmission by sending its own training symbol, followed by an amplified version of the signal received in MA phase. This training symbol facilitates TA and TB to perform self-interference suppression and to estimate the cascaded overall relay channel required for recovery of the data of interest. We derive lower bounds on the training-based individual rates and sum-rate of the two users and investigate the impact of channel estimation errors upon the sum-rate lower bound. Under the total transmit power constraint, we search for the optimal power allocation among the three nodes and optimal power allocation between data and training such that the sum-rate lower bound is maximized. Furthermore, we discuss the relationship between the optimal solutions and the relay location.