Many of the "counterintuitive" features of relativistic quantum field theory have their formal root in the Reeh-Schlieder theorem, which in particular entails that local operations applied to the vacuum state can produce any state of the entire field. It is of great interest, then, that I.E. Segal and, more recently, G. Fleming (in a paper entitled "Reeh-Schlieder Meets Newton-Wigner") have proposed an alternative "Newton-Wigner" localization scheme that avoids the Reeh-Schlieder theorem. In this paper, I reconstruct the Newton-Wigner localization scheme and clarify the limited extent to which it avoids the counterintuitive consequences of the Reeh-Schlieder theorem. I also argue that neither Segal nor Fleming has provided a coherent account of the physical meaning of Newton-Wigner localization.