Auditory attention decoding (AAD) through a brain-computer interface has had a flowering of developments since it was first introduced by Mesgarani and Chang (2012) using electrocorticograph recordings. AAD has been pursued for its potential application to hearing-aid design in which an attention-guided algorithm selects, from multiple competing acoustic sources, which should be enhanced for the listener and which should be suppressed. Traditionally, researchers have separated the AAD problem into two stages: reconstruction of a representation of the attended audio from neural signals, followed by determining the similarity between the candidate audio streams and the reconstruction. Here, we compare the traditional two-stage approach with a novel neural-network architecture that subsumes the explicit similarity step. We compare this new architecture against linear and non-linear (neural-network) baselines using both wet and dry electroencephalogram (EEG) systems. Our results indicate that the new architecture outperforms the baseline linear stimulus-reconstruction method, improving decoding accuracy from 66% to 81% using wet EEG and from 59% to 87% for dry EEG. Also of note was the finding that the dry EEG system can deliver comparable or even better results than the wet, despite the latter having one third as many EEG channels as the former. The 11-subject, wet-electrode AAD dataset for two competing, co-located talkers, the 11-subject, dry-electrode AAD dataset, and our software are available for further validation, experimentation, and modification.