Considering the separate management of transmission networks (TNs) and distribution networks (DNs), this article proposes a chance-constrained optimal power flow (CCOPF) formulation for integrated transmission and distribution (I-T&D) networks and its solution algorithm with limited information interaction. The uncertainties of loads and renewable generations are considered in the formulation, which guarantees that generations, power flows, and voltage magnitudes in both TNs and DNs remain within their bounds with a predefined probability. A double-iterative solution algorithm is proposed to solve CCOPF, of which the inner-iteration is applied to solve a deterministic OPF of I-T&D networks while the outer-iteration is to repeatedly update uncertain margins around a forecasted solution to converge finally. Particularly, the heterogeneous decomposition algorithm is applied to solve the deterministic OPF, and an IT&D-power-flow-based two-point estimation method is proposed to calculate uncertainty margins. The overall solution algorithm is realized based on boundary information exchange between TNs and DNs, where the data and model privacy of TNs and DNs are well-preserved. Numerical experiments demonstrate the accuracy and efficiency of the proposed solution algorithm.