This work presents the exact outage performance and throughput of two-way cognitive decode-and-forward relaying wireless sensor networks with realistic transceiver relay. The relay is a self-powered wireless node that harvests radio frequency energy from the transmitted signals. We consider four configurations of a network with formed by combining two bidirectional relaying protocols (multiple access broadcast protocol and time division broadcast protocol), and two power transfer policies (dual-source energy transfer and single-fixed-source energy transfer). Based on our analysis, we provide practical insights into the impact of transceiver hardware impairments on the network performance, such as the fundamental capacity ceiling of the network with various configurations that cannot be exceeded by increasing transmit power given a fixed transmission rate and the transceiver selection strategy for the network nodes that can optimize the implementation cost and performance tradeoff.