This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co-/quasi-differentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co- and quasidifferential under some natural assumptions on the integrand. Then we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the paper, we combine our results on the co-/quasi-differentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.