  We consider random q-state Potts models for $3\le q \le 8$ on the square
lattice where the ferromagnetic couplings take two values $J_1>J_2$ with equal
probabilities. For any q the model exhibits a continuous phase transition both
in the bulk and at the boundary. Using Monte Carlo techniques the surface and
the bulk magnetizations are studied close to the critical temperature and the
critical exponents $\beta_1$ and $\beta$ are determined. In the strip-like
geometry the critical magnetization profile is investigated with free-fixed
spin boundary condition and the characteristic scaling dimension,
$\beta_1/\nu$, is calculated from conformal field theory. The critical
exponents and scaling dimensions are found monotonously increasing with q.
Anomalous dimensions of the relevant scaling fields are estimated and the
multifractal behaviour at criticality is also analyzed.
