In this paper, we consider a denoise-and-forward (DNF) two-way relay network (TWRN) with non-coherent differential binary phase-shift keying modulation, where a battery-free relay node harvests energy from the received radio frequency (RF) signals and uses the harvested energy to help the source nodes for information exchange. Based on the power splitting (PS) and time switching (TS) receiver architectures, power splitting relaying (PSR) and time switching relaying (TSR) protocols at relay are studied. In order to investigate the effect of power allocations on two source nodes, power splitting coefficient and time switching factor at relay on performance, the two proposed protocols are analyzed and the bit error rate (BER) expressions of end-to-end system are derived. Based on these expressions, the optimal power of sources, the power splitting ratio and the time switching factor are obtained via the numerical search method. The simulation and numerical results provide practical insights into the effect of various system parameters, such as the power splitting coefficient, the time switching factor, sources to relay distances, the noise power, and the energy harvesting efficiency on the performance of this TWRN. In addition, the results show that the PSR protocol outperforms the TSR protocol in terms of throughput under various network geometries.