  Universal properties of the statistics of stochastic efficiency for
mesoscopic time-reversal symmetry broken energy transducers are revealed in the
Gaussian approximation. We also discuss how the second law of thermodynamics
restricts the statistics of stochastic efficiency. The tight-coupling
(reversible) limit becomes unfavorable, characterized by an infinitely broad
distribution of efficiency at {\em all times}, when time-reversal symmetry
breaking leads to an asymmetric Onsager response matrix. The underlying physics
is demonstrated through the integer quantum Hall effect and further elaborated
in a triple-quantum-dot three-terminal thermoelectric engine.
