We theoretically investigate schemes to discriminate between two nonorthogonal quantum states given multiple copies. We consider a number of state discrimination schemes as applied to nonorthogonal, mixed states of a qubit. In particular, we examine the difference that local and global optimization of local measurements makes to the probability of obtaining an erroneous result, in the regime of finite numbers of copies N, and in the asymptotic limit as N→∞. Five schemes are considered: optimal collective measurements over all copies, locally optimal local measurements in a fixed single-qubit measurement basis, globally optimal fixed local measurements, locally optimal adaptive local measurements, and globally optimal adaptive local measurements. Here an adaptive measurement is one in which the measurement basis can depend on prior measurement results. For each of these measurement schemes we determine the probability of error (for finite N) and the scaling of this error in the asymptotic limit. In the asymptotic limit, it is known analytically (and we verify numerically) that adaptive schemes have no advantage over the optimal fixed local scheme. Here we show moreover that, in this limit, the most naive scheme (locally optimal fixed local measurements) is as good as any noncollective scheme except for states with less than 2% mixture. For finite N, however, the most sophisticated local scheme (globally optimal adaptive local measurements) is better than any other noncollective scheme for any degree of mixture.