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. In this work, 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 wet and dry systems can deliver comparable results despite the latter having one third as many EEG channels as the former, and that the new architecture outperforms the baseline stimulus-reconstruction methods for both EEG modalities. The 14-subject, wet-electrode AAD dataset for two competing, co-located talkers, the 11-subject, dry-electrode AAD dataset, and our software are available to download for further validation, experimentation, and modification.