A unique compact Poisson neuron that encodes information in the tunable duty cycle of probabilistic spike trains is presented as an enabling technology for cost-effective spiking neural network (SNN) hardware. The Poisson neuron exploits the back-hopping oscillation (BHO) in scalable spin-transfer torque (STT)-MRAM. The macrospin LLGS simulation confirms that the coupled local Joule heating and STT effects are responsible for the bias-dependent BHO. The complete neuron circuit design is at least 6*smaller than the state-of-the-art integrate-and- fire (IF) CMOS neuron. Hardware-friendly all-spin deep SNNs achieve equivalent accuracy to deep neural networks (DNN), 98.4 % for MNIST, even when considering the probabilistic nature of neurons.