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Variational ab initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows, in principle, straightforward extraction of any other observable of interest, besides the energy, but, in practice, this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning Ansätze to obtain highly accurate wave functions free of basis set errors and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties. [ABSTRACT FROM AUTHOR]