  Young, rapidly spinning magnetars are invoked as central engines behind a
diverse set of transient astrophysical phenomena, including gamma-ray bursts
(GRB), super-luminous supernovae (SLSNe), fast radio bursts (FRB), and binary
neutron star (NS) mergers. However, a barrier to direct confirmation of the
magnetar hypothesis is the challenge of directly observing non-thermal emission
from the central engine at early times (when it is most powerful and thus
detectable) due to the dense surrounding ejecta. We present CLOUDY calculations
of the time-dependent evolution of the temperature and ionization structure of
expanding supernova or merger ejecta due to photo-ionization by a magnetar
engine, in order to study the escape of X-rays (absorbed by neutral gas) and
radio waves (absorbed by ionized gas), as well as to assess the evolution of
the local dispersion measure due to photo-ionization. We find that ionization
breakout does not occur if the engine's ionizing luminosity decays rapidly, and
that X-rays typically escape the oxygen-rich ejecta of SLSNe only on $\sim 100
\, {\rm yr}$ timescales, consistent with current X-ray non-detections. We apply
these results to constrain engine-driven models for the binary NS merger
GW170817 and the luminous transient ASASSN-15lh. In terms of radio transparency
and dispersion measure constraints, the repeating FRB 121102 is consistent with
originating from a young, $\gtrsim 30-100 \, {\rm yr}$, magnetar similar to
those inferred to power SLSNe. We further show that its high rotation measure
can be produced within the same nebula that is proposed to power the quiescent
radio source observed co-located with FRB 121102. Our results strengthen
previous work suggesting that at least some FRBs may be produced by young
magnetars, and motivate further study of engine powered transients.
