Data from the NASA Magnetospheric Multiscale mission are used to investigate asymmetric magnetic reconnection at the dayside boundary between the Earth's magnetosphere and the solar wind. High‐resolution measurements of plasmas and fields are used to identify highly localized (~15 electron Debye lengths) standing wave structures with large electric field amplitudes (up to 100 mV/m). These wave structures are associated with spatially oscillatory energy conversion, which appears as alternatingly positive and negative values of J · E. For small guide magnetic fields the wave structures occur in the electron stagnation region at the magnetosphere edge of the electron diffusion region. For larger guide fields the structures also occur near the reconnection X‐line. This difference is explained in terms of channels for the out‐of‐plane current (agyrotropic electrons at the stagnation point and guide field‐aligned electrons at the X‐line). Key Points Energy conversion is highly localized within asymmetric reconnection electron diffusion regions Oscillatory reconnection electric fields show characteristics of both spatial structures and propagating waves that are consistent with standing oblique quasi‐electrostatic whistlers Both positive and negative values of J · E result from uniform current and oscillating electric fields