Pulsed-electric drilling systems can be augmented with multi-component electromagnetic field sensors on the drillstring, at the earth's surface, or in existing boreholes in the vicinity of the planned drilling path. The sensors detect electrical fields and/or magnetic fields caused by the electrical pulses and derive therefrom information of interest including, e.g., spark size and orientation, bit position, at-bit resistivity and permittivity, and tomographically mapped formation structures. The at-bit resistivity measurements can be for anisotropic or isotropic formations, and in the former case, can include vertical and horizontal resistivities and an orientation of the anisotropy axis. The sensors can illustratively include toroids, electrode arrays, tilted coil antennas, magnetic dipole antennas aligned with the tool axes, and magnetometers. The use of multiple such sensors increases measurement accuracy and the number of unknown model parameters which can be derived using an iterative inversion technique.