Chest X-rays (CXRs) represent the first tool globally employed to detect cardiopulmonary pathologies. These acquisitions are highly affected by scattered photons due to the large field of view required. Scatter in CXRs introduces background in the images, which reduces their contrast. We developed three deep-learning-based models to estimate and correct scatter contribution to CXRs. We used a Monte Carlo (MC) ray-tracing model to simulate CXRs from human models obtained from CT scans using different configurations (depending on the availability of dual-energy acquisitions). The simulated CXRs contained the separated contribution of direct and scattered X-rays in the detector. These simulated datasets were then used as the reference for the supervised training of several NNs. Three NN models (single and dual energy) were trained with the MultiResUNet architecture. The performance of the NN models was evaluated on CXRs obtained, with an MC code, from chest CT scans of patients affected by COVID-19. The results show that the NN models were able to estimate and correct the scatter contribution to CXRs with an error of <5%, being robust to variations in the simulation setup and improving contrast in soft tissue. The single-energy model was tested on real CXRs, providing robust estimations of the scatter-corrected CXRs.