Carpooling is an integral component in smart carbon-neutral cities, in particular to facilitate homework commuting. We study an innovative carpooling service developed by the start-up Ecov which specialises in homework commutes in peri-urban and rural regions. When a passenger makes a carpooling request, a designated driver is not assigned as in a traditional carpooling service; rather the passenger waits for the first driver, from a population of non-professional drivers who are already en route, to arrive. We propose a two-stage Bayesian hierarchical model to overcome the considerable difficulties, due to the sparsely observed driver and passenger data from an embryonic stochastic carpooling service, to deliver high-quality predictions of driver flow and passenger waiting times. The first stage focuses on the driver flow, whose predictions are aggregated at the daily level to compensate the data sparsity. The second stage processes this single daily driver flow into sub-daily (e.g. hourly) predictions of the passenger waiting times. We demonstrate that our model mostly outperforms frequentist and non-hierarchical Bayesian methods for observed data from operational carpooling service in Lyon, France and we also validated our model on simulated data.