Quantum state discrimination is a fundamental problem in quantum information science. We investigate the optimal distinguishability of orthogonal two-qubit (bipartite) quantum states. The scenario consists of three parties: Alice, Bob, and Charlie. Charlie prepares one of two orthogonal states and sends one qubit to Alice and the other to Bob. Their goal is to correctly identify which state Charlie sent. In most state discrimination scenarios, it is assumed that Alice and Bob can freely communicate with one another so as to collectively agree on the best guess. In this research, we consider a more restricted setting where only one-way classical communication is possible from Alice to Bob. Under this setting, we study two figures of merit (i) Alice's optimal probability, $P$, of identifying the state , and (ii) Alice's optimal probability, $P^\perp$, of identifying the state along with helping Bob identify the state perfectly. We show that in general $P\neq P^\perp$ and we prove a theorem for when $P=P^\perp$. We also found that the maximum of $P-P^\perp$ can arbitrarily approach $1/2$.