In this paper, we introduce a data-driven methodology for estimating capacity regions of continuous channels in multi-user communication systems. Computing capacity regions is a long standing open problem, even in simple communication scenarios. Nevertheless, it is often possible to represent their capacity regions as the limit of an optimization problem (a multi-letter expression). In many cases, these multi-letter expressions can be expressed in terms of directed information (DI) rates. Accordingly, our approach utilizes neural networks to estimate capacity regions, leveraging the recent introduction of the directed information neural estimator (DINE). The main idea of our methodology involves training DINE-based models using samples of channel inputs and outputs, and using these models to estimate the DI rate terms that are intrinsic to the studied capacity region. To estimate the capacity region rates, we optimize the DI rates over the involved input distributions which are parameterized by a neural distribution transformer (NDT), and execute an alternating maximization procedure between the NDT models and DINE-based models until convergence is achieved. The methodology is suitable for the case where the channel is treated as a "black-box"and the designer can only gather observations of its inputs and outputs, lacking any knowledge of the explicit channel model. The performance of our proposed algorithm is shown via several well-known settings, including the Gaussian two-way channel and the two-user Gaussian multiple-access channel with and without feedback.