We discuss avalanche and finite size fluctuations in a mesoscopic model to describe the shear plasticity of amorphous materials. Plastic deformation is assumed to occur through series of local reorganizations. Yield stress criteria are random while each plastic slip event induces a quadrupolar long range elastic stress redistribution. The model is discretized on a regular square lattice. Shear plasticity can be studied in this context as a depinning dynamic phase transition. We show evidence for a scale free distribution of avalanches $P(s)\propto S^{-\kappa}$ with a non trivial exponent $\kappa \approx 1.25$ significantly different from the mean field result $\kappa = 1.5$. Finite size effects allow for a characterization of the scaling invariance of the yield stress fluctuations observed in small samples. We finally identify a population of precursors of plastic activity and characterize its spatial distribution.