We study nanopillar spin-torque oscillators processed from low-resistance-area product MgO-based magnetic tunnel junctions. The influence of spin torque can be seen in quasistatic experiments as a strong astroid distortion, consistent with a pure Slonczewski-type spin torque. At microwave frequencies, the spin-torque results in pronounced magnetization auto-oscillations with a clear threshold behavior, though only evidenced in the antiparallel configuration. Two kinds of auto-oscillations are seen depending on the polarity of the voltage applied to the junction. Free layer oscillations require a large easy axis applied field and electrons flowing from the reference layers to the free layer. Acoustic excitation of the reference synthetic ferrimagnet can be seen, provided the electron flow is reversed, indicating a clear similarity with the behavior of conventional all-metallic spin valves. The analysis of the threshold indicates that the amplitude of the spin torque is neither proportional to the current nor to the voltage and it is not accompanied by any significant fieldlike torque.