In this paper, we study a training design problem for multiple-inputmultiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) amplify-and-forward (AF) two-way relay networks. Unlike the existing studies, we assume the spatially correlated fading and consider the nonreciprocal channel condition, which is a more practical assumption but makes the training problem more challenging. The equivalent channels of bidirectional relaying links, which consist of self-interfering channels and information-bearing channels, are estimated at each source node based on a linear minimum mean square error (LMMSE) approach. The total mean square error (MSE) of the channel estimation is minimized under the transmit power constraints at the source nodes and at the relay. To solve this problem, we first derive an optimal structure of the training signals, and then, convert the optimization problem into a tractable convex form, from which the optimal training scheme is designed efficiently. Furthermore, for a practical special case, the optimal training design is derived in semi-closed form, which provides useful insights. To reduce the required complexity, a low-complexity training scheme is also derived in closed-form. This scheme is shown to be asymptotically optimal in the high signal-to-noise ratio (SNR) regime and gives further insights into the optimal training. The performance of the proposed schemes is demonstrated through numerical simulations.