We study electron transport through single-electron spin-valve transistors in the presence of nonlocal exchange between the ferromagnetic leads and the central normal-metal island. The Coulomb interaction is described with the "orthodox model" for Coulomb blockade and we allow for noncollinear lead magnetization directions. Two distinct exchange mechanisms that have been discussed in the literature are shown to be of comparable strength and are taken into account on equal footing. We present results for the linear conductance as a function of gate voltage and magnetic configuration, and discuss the response of the system to applied magnetic fields.