Using relativistic, axisymmetric, ideal MHD, we examine the outflow from a debris disk around a newly formed stellar-mass black hole, taking into account the baryonic matter, the electron-positron/photon fluid, and the large-scale electromagnetic field. We clarify the relationship between the thermal (fireball) and magnetic (Poynting flux) acceleration mechanisms, identify the parameter regimes where qualitatively different behaviors are expected, and demonstrate that the observationally inferred properties of the GRB outflows can be attributed to magnetic driving. We show that the Lorentz force can convert up to 50% of the initial total energy into kinetic energy of a collimated flow of baryons. This energy, in turn, may be converted into radiation by internal shocks. We examine how baryon loading and magnetic collimation affect the structure of the flow.