In the first part of the paper we develop first the sensitivity analysis for the nonlinear McKean-Vlasov diffusions stressing precise estimates of growth of the solutions and their derivatives with respect to the initial data, under rather general assumptions on the coefficients. The exact estimates become particularly important when treating the extension of these equations having random coefficient, since the noise is usually assumed to be unbounded. The second part contains our main results dealing with the sensitivity of stochastic McKean-Vlasov diffusions. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the estimates obtained in the first part. The motivation for studying sensitivity of McKean-Vlasov SDEs arises naturally from the analysis of the mean-field games with common noise.