An explanation is given for previous numerical results which suggest a certain bifurcation of `vector solitons' from scalar (single-component) solitary waves in coupled nonlinear Schrödinger (CNLS) systems. The bifurcation in question is nonlocal in the sense that the vector soliton does not have a small-amplitude component, but instead approaches a solitary wave of one component with two infinitely far-separated waves in the other component. Yet, it is argued that this highly nonlocal event can be predicted from a purely local analysis of the central solitary wave alone. Specifically, the linearization around the central wave should contain asymptotics which grow at precisely the speed of the other-component solitary waves on the two wings. This approximate argument is supported by both a detailed analysis based on matched asymptotic expansions, and numerical experiments on two example systems. The first is the usual CNLS system involving an arbitrary ratio between the self-phase and cross-phase-modulation terms, and the second is a CNLS system with saturable nonlinearity that has recently been demonstrated to support stable multi-peaked solitary waves. The asymptotic analysis further reveals that when the curves which define the proposed criterion for scalar nonlocal bifurcations intersect with boundaries of certain local bifurcations, the nonlocal bifurcation could turn from scalar to nonscalar at the intersection. This phenomenon is observed in the first example. Lastly, we have also selectively tested the linear stability of several solitary waves just born out of scalar nonlocal bifurcations. We found that they are linearly unstable. However, they can lead to stable solitary waves through parameter continuation.