An experimental study of the statistical distribution of acoustic emission avalanches in martensitic transitions is presented. The critical exponents characterizing the power-law distributions of avalanches are measured as a function of cycling under soft-driving (stress-driving) and hard-driving (strain-driving) procedures. Results provide experimental support for a recent theory based on a spin model that predicts a crossover from classical criticality in the soft-driving case to self-organized criticality in the hard-driving case.