  Given a surface $M$ and a Borel probability measure $\nu$ on the group of
$C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on
$M$. We prove for hyperbolic stationary measures the following trichotomy:
either the stable distributions are non-random, the measure is SRB, or the
measure is supported on a finite set and is hence almost-surely invariant. In
the proof of the above results, we study skew products with surface fibers over
a measure preserving transformations equipped with a decreasing
sub-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number
of applications of our main theorem are presented.
