  We study temporal entanglement in dual-unitary Clifford circuits with
probabilistic measurements preserving spatial unitarity. We exactly
characterize the temporal entanglement barrier in the measurement-free regime,
exhibiting ballistic growth and decay and a volume-law peak. In the presence of
measurements, we relate the temporal entanglement to the scrambling properties
of the circuit. For "good scramblers" measurements do not qualitatively change
the temporal entanglement profile but only result in a reduced entanglement
velocity, whereas for "poor scramblers" the initial ballistic growth of
temporal entanglement with bath size is modified to diffusive. This difference
is understood through a mapping of the underlying operator dynamics to a biased
and an unbiased persistent random walk respectively. In the latter case
measurements additionally modify the ballistic decay to the perfect dephaser
limit, with vanishing temporal entanglement, to an exponential decay, which we
describe through a spatial transfer matrix method. This spatial dynamics is
shown to be described by a non-Hermitian hopping model, exhibiting a
PT-breaking transition at a critical measurement rate $p=1/2$. In all cases the
peak value of the temporal entanglement barrier exhibits volume-law scaling for
all measurement rates.
