We study the properties of strain bursts (dislocation avalanches) occurring in two-dimensional discrete dislocation dynamics models under quasistatic stress-controlled loading. Contrary to previous suggestions, the avalanche statistics differ fundamentally from predictions obtained for the depinning of elastic manifolds in quenched random media. Instead, we find an exponent τ=1 of the power-law distribution of slip or released energy, with a cutoff that increases exponentially with the applied stress and diverges with system size at all stresses. These observations demonstrate that the avalanche dynamics of 2D dislocation systems is scale-free at every applied stress and, therefore, cannot be envisaged in terms of critical behavior associated with a depinning transition.