In a recent work [Pallister, Linden, and Montanaro, Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify nonmaximally entangled two-qubit pure states under the constraint that the accessible measurements are locally projective and nonadaptive. Their good result leads naturally to the following question: What is the optimal strategy among general local operations and classical communication (LOCC) measurements? In this paper, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical communication (one-way LOCC) measurements; (ii) local operations and two-way classical communication (two-way LOCC) measurements; and (iii) separable measurements. Surprisingly, our results reveal that for the two-qubit pure state verification problem two-way LOCC measurements remarkably outperform one-way LOCC measurements and have the same power as the separable measurements.