Training schemes for full duplex two-way relays are investigated. We propose a novel one-block training scheme with a maximum likelihood estimator to estimate the channels between the nodes as well as the residual self-interference (RSI) channel simultaneously. A quasi-Newton algorithm is used to solve the estimator. As a baseline, a multi-block training scheme is also considered. The Cramer-Rao bounds of the one-block and multi-block training schemes are derived. By using the Szegö's theorem about Toeplitz matrices, we analyze how the channel parameters and transmit powers affect the Fisher information. We show analytically that exploiting the structure arising from the RSI channel increases its Fisher information. Numerical results show the benefits of estimating the RSI channel.