In this paper, we address a two-way relay network (TWRN) where two source nodes exchange their information through a relay node (RN) in a bi-directional manner and propose a receiver design method at RN and source nodes under variable-gain amplify-and-forward (VG-AF) relay system. A two-period transmission protocol is employed: in the training period, source node and RN will send their own training signals to obtain channel state information (CSI); in the transmission period, two source nodes exchange their information based on the CSI obtained in the training period. We develop the Maximum-Likelihood (ML) estimator and Linear-Minimum-Mean-Square-Errors (LMMSE) estimator and derive the expression of channel estimation errors. Numerical results show that VG relay system outperforms fixed gain (FG) system in terms of mean square errors (MSE) of channel estimation and bit error rate (BER) performance. Besides, in contrast to the high Peak-to-Average-Power-Ratio (PAPR) in FG relay system, simulation results show that VG relay system has a more stable transmission power.