This paper investigates the problem of distributed convex optimisation under constrained communication. A novel stochastic event-triggering algorithm is shown to solve the problem asymptotically to any arbitrarily small error without exhibiting Zeno behaviour. A systematic design of the stochastic event processes is then derived from the analysis on optimality and communication rate with the help of a meta-optimisation problem. Lastly, a numerical example on distributed classification is provided to visualise the performance of the proposed algorithm in terms of convergence in optimisation error and average communication rate with comparison to other algorithms in the literature. We show that the proposed algorithm is highly effective in reducing communication rates compared with algorithms proposed in the literature.