The Local Group is a unique environment in which to study the astrophysics of galaxy formation. The proximity of the Milky Way and M31 causes a large fraction of the low-mass halo population to interact 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 halo that passed through the haloes of both the Milky Way and M31 at early times and is almost twice as concentrated as isolated field haloes. These Hermeian haloes are distributed anisotropically at greater distances from the Local Group barycentre than the primary haloes and appear to cluster close to the Milky Way and M31 in projection. We show that some Hermeian haloes can host galaxies that are promising targets for indirect dark matter searches and are competitive with signals from other dwarf galaxies. Hermeian galaxies in the Local Group should be detectable by forthcoming wide-field imaging surveys.