In the local Universe, there is a strong division in the star-forming properties of low-mass galaxies, with star formation largely ubiquitous amongst the field population while satellite systems are predominantly quenched. This dichotomy implies that environmental processes play the dominant role in suppressing star formation within this low-mass regime (${M}_{\star} \sim 10^{5.5-8}~{\rm M}_{\odot}$)... As shown by observations of the Local Volume, however, there is a non-negligible population of passive systems in the field, which challenges our understanding of quenching at low masses. By applying the satellite quenching models of Fillingham et al. (2015) to subhalo populations in the Exploring the Local Volume In Simulations (ELVIS) suite, we investigate the role of environmental processes in quenching star formation within the nearby field. Using model parameters that reproduce the satellite quenched fraction in the Local Group, we predict a quenched fraction -- due solely to environmental effects -- of $\sim 0.52 \pm 0.26$ within $1< R/R_{\rm vir} < 2$ of the Milky Way and M31. This is in good agreement with current observations of the Local Volume and suggests that the majority of the passive field systems observed at these distances are quenched via environmental mechanisms. Beyond $2~R_{\rm vir}$, however, dwarf galaxy quenching becomes difficult to explain through an interaction with either the Milky Way or M31, such that more isolated, field dwarfs may be self-quenched as a result of star-formation feedback. (read more)