We consider the amount of energy dissipated during individual avalanches at the depinning transition of disordered and athermal elastic systems. Analytical progress is possible in the case of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model for Barkhausen noise, due to an exact mapping between the energy released in an avalanche and the area below a Brownian path until its first zero-crossing. Scaling arguments and examination of an extended mean-field model with internal structure show that dissipation relates to a critical exponent recently found in a study of the rounding of the depinning transition in the presence of activated dynamics. A new numerical method to compute the dynamic exponent at depinning in terms of blocked and marginally stable configurations is proposed, and a kind of "dissipative anomaly" ¿with potentially important consequences for non-equilibrium statistical mechanics¿ is discussed. We conclude that for depinning systems the size of an avalanche does not constitute by itself a univocal measure of the energy dissipated.