  The class of stochastically self-similar sets contains many famous examples
of random sets, e.g. Mandelbrot percolation and general fractal percolation.
Under the assumption of the uniform open set condition and some mild
assumptions on the iterated function systems used, we show that the
quasi-Assouad dimension of self-similar random recursive sets is almost surely
equal to the almost sure Hausdorff dimension of the set. We further comment on
random homogeneous and $V$-variable sets and the removal of overlap conditions.
