This paper designs two energy-efficient full-duplex transmission strategies to maximize energy efficiency (EE) of two-way amplify-and-forward relay (TWAFR) networks with simultaneous wireless information and power transfer. In the two designed transmission strategies, there are direct links (DLs) between two end nodes and the DLs can be used to convey more information. At the same time, one of the end nodes is considered battery limited and has the capability of energy harvesting from the received signals. With the first designed transmission strategy, the EE can be maximized by optimizing the power allocation, and the analytical expressions of optimal power allocation (OPA) are obtained with transformation of EE maximization problem. With the second transmission strategy, the EE can also be maximized by optimizing the power allocation, and the numerical solutions of OPA are obtained with alternating optimal algorithm. Simulations show that the EEs of the TWAFR networks can be improved with our two designed transmission strategies.