  We analyse the quenched fractions, gas content, and star formation histories
of ~1200 satellite galaxies with $M_* \geq 5 \times 10^6~{\rm M}_\odot$ around
198 Milky Way- (MW) and Andromeda-like (M31) hosts in TNG50, the
highest-resolution simulation of IllustrisTNG. Satellite quenched fractions are
larger for smaller masses, for smaller distances to their host galaxy, and in
the more massive M31-like compared to MW-like hosts. As satellites cross their
host's virial radius, their gas content drops: most satellites within 300 kpc
lack detectable gas reservoirs at $z=0$, unless they are massive like the
Magellanic Clouds and M32. Nevertheless, their stellar assembly exhibits a
large degree of diversity. On average, the cumulative star formation histories
are more extended for brighter, more massive satellites with a later infall,
and for those in less massive hosts. Based on these relationships, we can even
infer infall periods for observedMWand M31 dwarfs: e.g. 0-4 Gyr ago for the
Magellanic Clouds and Leo I, 4-8 and 0-2 Gyr ago for M32 and IC 10,
respectively. Ram pressure stripping (in combination with tidal stripping)
deprives TNG50 satellites of their gas reservoirs and ultimately quenches their
star formation, even though only a few per cent of the present-day satellites
around the 198 TNG50 MW/M31-like hosts appear as jellyfish. The typical time
since quenching for currently quenched TNG50 satellites is
$6.9^{+2.5}_{-3.3}~{\rm Gyr}$ ago. The TNG50 results are consistent with the
quenched fractions and stellar assembly of observed MW and M31 satellites,
however, satellites of the SAGA survey with $M_* \sim 10^{8-9}~{\rm M}_\odot$
exhibit lower quenched fractions than TNG50 and other, observed analogues.
