Printed electronics (PE) offers flexible, extremely low-cost, and on-demand hardware due to its additive manufacturing process, enabling emerging ultra-low-cost applications, including machine learning applications. However, large feature sizes in PE limit the complexity of a machine learning classifier (e.g., a neural network (NN)) in PE. Stochastic computing Neural Networks (SC-NNs) can reduce area in silicon technologies, but still require complex designs due to unique implementation tradeoffs in PE. In this paper, we propose a printed mixed-signal system, which substitutes complex and power-hungry conventional stochastic computing (SC) components by printed analog designs. The printed mixed-signal SC consumes only 35% of power consumption and requires only 25% of area compared to a conventional 4-bit NN implementation. We also show that the proposed mixed-signal SC-NN provides good accuracy for popular neural network classification problems. We consider this work as an important step towards the realization of printed SC-NN hardware for near-sensor-processing.