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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 FIRE-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 less than 500 Myr. However, if we select hosts with an LMC-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 about 1 sigma 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 LCDM cosmology.