  We predict and compare the distributions and properties of hyper-velocity
stars (HVSs) ejected from the centres of the Milky Way (MW) and the Large
Magellanic Cloud (LMC). In our model, HVSs are ejected at a constant rate --
equal in both galaxies -- via the Hills mechanism and are propagated in a
combined potential, where the LMC orbits the MW on its first infall. By
selecting $m>2\, \mathrm{M_\odot}$ HVSs well-separated from the Magellanic
Clouds and Galactic midplane, we identify mock HVSs which would stand out from
ordinary stars in the stellar halo in future data releases from the Gaia
satellite and the Vera C. Rubin Observatory's Legacy Survey of Space and Time
(LSST). We find that in these deep surveys, LMC HVSs will outnumber MW ones by
a factor $\sim 2.5$, as HVSs can more easily escape from the shallower
potential of the LMC. At an assumed HVS ejection rate of $10^{-4} \,
\mathrm{yr^{-1}}$, HVSs detectable in the final Gaia data release and LSST from
the LMC (MW) will number $125_{-12}^{+11}$ ($50_{-8}^{+7}$) and
$140_{-11}^{+10}$ ($42_{-7}^{+6}$), respectively. The MW and LMC HVS
populations show different kinematics and spatial distributions. While LMC HVSs
have more modest total velocities and larger Galactocentric distances clustered
around those of the LMC itself, HVSs from the MW show broader distributions,
including a prominent high-velocity tail above $500 \, \mathrm{km \ s^{-1}}$
that contains at least half of the stars. These predictions are robust against
reasonable variation of the Galactic potential and of the LMC central black
hole mass.
