<p>The <i>weighted vertex cover</i> problem revolves around selecting a subset of vertices that covers a target edge set while minimizing the total cost of the selected vertices. We consider a variant of this classic optimization problem where the target edge set is not fully known; rather, it is characterized by a probability distribution. Adhering to the model of <i>two-stage stochastic optimization</i>, the execution is divided into two stages. In the first stage, the decision maker selects a vertex subset based on the probabilistic forecast of the target edge set. In the second stage, the target edge set is revealed, and the decision maker can augment the initial vertex subset with additional vertices to ensure coverage; however, this augmentation is more expensive due to increased vertex costs. This paper initiates the study of the two-stage stochastic vertex cover problem in the realm of <i>distributed graph algorithms</i>, where the decision-making process is distributed among the graph’s vertices. We consider two known stochastic optimization variants: the <i>independent sampling</i> model, where the edges in the target set are drawn independently from some probability distribution; and the <i>finite scenario</i> model, where the probability distribution over the target edge set is provided explicitly. For both variants, we devise efficient distributed algorithms based on a novel adaptation of the distributed primal-dual technique to linear programs resulting from the stochastic optimization problems’ relaxation.</p>