  What the progenitors of fast radio bursts (FRBs) are, and whether there are
multiple types of progenitors are open questions. The advent of localized FRBs
with host galaxy redshifts allows the various emission models to be directly
tested for the first time. Given the recent localizations of two non-repeating
FRBs (FRB 180924 and FRB 190523), we discuss a selection of FRB emission models
and demonstrate how we can place constraints on key model parameters like the
magnetic field strength and age of the putative FRB-emitting neutron star. In
particular, we focus on models related to compact binary merger events
involving at least one neutron star, motivated by commonalities between the
host galaxies of the FRBs and the hosts of such merger events/short gamma-ray
bursts (SGRBs). We rule out the possibility that either FRB was produced during
the final inspiral stage of a merging binary system. Where possible, we predict
the light curve of electromagnetic emission associated with a given model and
use it to recommend multi-wavelength follow-up strategies that may help confirm
or rule out models for future FRBs. In addition, we conduct a targeted
sub-threshold search in Fermi Gamma-ray Burst Monitor data for potential SGRB
candidates associated with either FRB, and show what a non-detection means for
relevant models. The methodology presented in this study may be easily applied
to future localized FRBs, and adapted to sources with possibly core-collapse
supernova progenitors, to help constrain potential models for the FRB
population at large.
