The Local Group is a unique environment in which to study the astrophysics of galaxy formation. The proximity of the Milky Way and M31 enhances the frequency of interactions of the low-mass halo population with more massive dark matter haloes, which increases their concentrations and strips them of gas and other material. Some low-mass haloes pass through the haloes of the Milky Way or M31 and are either ejected into the field or exchanged between the two primary hosts. We use high resolution gas-dynamical simulations to describe a new class of field haloes that passed through the haloes of both the Milky Way and M31 at early times and are almost twice as concentrated as field haloes that do not interact with the primary pair. These ‘Hermeian’ haloes are distributed anisotropically at larger distances from the Local Group barycentre than the primary haloes and appear to cluster along the line connecting the Milky Way and M31. Hermeian haloes facilitate the exchange of dark matter, gas, and stars between the Milky Way and M31 and can enhance the star formation rates of the gas in the primary haloes during their interactions with them. We also show that some Hermeian haloes can host galaxies that, because they are embedded in haloes that are more concentrated than regular field haloes, are promising targets for indirect dark matter searches beyond the Milky Way virial radius and can produce signals that are competitive with those of some dwarf galaxies. Hermeian galaxies in the Local Group should be detectable by forthcoming wide-field imaging surveys.