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We study distributed algorithms for adapting transmit beamformers and linear receiver filters in a Time-Division Duplex Multiple-Input Multiple-Output (MIMO) interference network. Each transmitter transmits a single beam, and neither the transmitters nor receivers have a priori Channel State Information (CSI). Given a fixed set of powers, we present an adaptive version of the Max-SINR algorithm: pilot symbols are alternately transmitted in the forward direction (transmitters to receivers) and in the reverse direction (receivers to transmitters). Unlike previous channel estimation schemes, transmissions in each direction are synchronized across the source or destination nodes, and the pilots are used to update the filters/beams directly using a least squares criterion. To improve the performance with limited training, we include exponential weighting of the least squares objective across data frames. In addition, bi-directional training can be used to implement analog interference pricing for power control: training in the forward direction is used to measure received signal-to-interference plus noise ratios (SINRs) and interference prices, and those estimates combined with synchronous backward training are used to update the powers. Given sufficient training this method achieves the same performance as interference pricing updates with perfect CSI. Numerical results are presented that illustrate the performance of these methods in different settings.