We examine the prevalence, longevity, and causes of planes of satellite dwarf galaxies, as observed in the Local Group. We use 14 Milky Way/Andromeda-(MW/M31) mass host galaxies from the Feedback In Realistic Environments-2 simulations. We select the 14 most massive satellites by stellar mass within ≤ 300, kpc of each host and correct for incompleteness from the foreground galactic disc when comparing to the MW. We find that MW-like planes as spatially thin and/or kinematically coherent as observed are uncommon, but they do exist in our simulations. Spatially thin planes occur in 1-2 per cent of snapshots during z = 0-0.2, and kinematically coherent planes occur in 5 per cent of snapshots. These planes are generally transient, surviving for <500 Myr. However, if we select hosts with a Large Magellanic Cloud-like satellite near first pericentre, the fraction of snapshots with MW-like planes increases dramatically to 7-16 per cent, with lifetimes of 0.7-1 Gyr, likely because of group accretion of satellites. We find that M31's satellite distribution is much more common: M31's satellites lie within ∼1σ of the simulation median for every plane metric we consider. We find no significant difference in average satellite planarity for isolated hosts versus hosts in LG-like pairs. Baryonic and dark matter-only simulations exhibit similar levels of planarity, even though baryonic subhaloes are less centrally concentrated within their host haloes. We conclude that planes of satellites are not a strong challenge to ΛCDM cosmology.