ABSTRACT 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 such as e 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 multiwavelength 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.