For problems when decisions are taken prior to observing the realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern. A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framework that includes and extends various existing formulations. The unifying approach is built on the basis of a recent generation of bundle methods called with on-demand accuracy, characterized by its versatility and flexibility. Numerical results for several difficult probabilistically constrained problems confirm the interest of the approach.