Microgrids (MGs) have attracted growing attention due to self-sufficiency and self-healing properties. Nonetheless, the intermittent nature and uncertainty of distributed energy resources and load demands remain challenging issues in balancing demands and managing energy resources in MGs. Existing research efforts mainly focus on developing techniques to enable interactions between local MGs and the utility grid, which leads to high line power losses and operation costs. In this paper, we present the Sto2Auc framework to address the issue of stochastic optimal bidding problem for a system with MGs. First, the optimal bidding problem is formulated as a two-stage stochastic programming process, which aims to minimize the system operation cost and obtain optimal energy capacity of MGs by the MG center controller (MGCC). Uncertainties arise from both energy supply and demand, which are considered in the stochastic model, and random parameters representing those uncertainties are captured by using the Monte Carlo method. Second, to enable optimal electricity trading between the insufficient and surplus MGs, we propose a distributed double auction (DDA)-based scheme, which is proven to converge to the optimal social welfare of the system with MGs, and achieves the economical properties of being strategy-proof, individually rational, and (weak) budget balanced. Extensive experiments on an MG system composed of IEEE-33 buses demonstrate the effectiveness of proposed scheme. The experimental results show that Sto2Auc framework is capable of reducing the operational cost of MG systems, while the implemented DDA scheme achieves good performance with respect to social welfare, demand insufficiency, and MGCC profit.