The exchange of information is a crucial factor in achieving consensus among agents. However, in real-world scenarios, nonideal information sharing is prevalent due to complex environmental conditions. Consider the information distortions (data) and stochastic information flow (media) during state transmission both caused by physical constraints, a novel model of transmission-constrained consensus over random networks is proposed in this work. The transmission constraints are represented by heterogeneous functions that reflect the impact of environmental interference in multiagent systems or social networks. A directed random graph is applied to model the stochastic information flow where every edge is connected probabilistically. Using stochastic stability theory and the martingale convergence theorem, it is demonstrated that the agent states will converge to a consensus value with probability 1, despite information distortions and randomness in information flow. Numerical simulations are presented to validate the effectiveness of the proposed model.