We present an exhaustive methodology for fitting Compton-thick X-ray reprocessor models to obscured active galactic nuclei (AGNs) and for interpreting the results. We focus on the mytorus model but also include some analysis from other models. The models are applied specifically to Suzaku, BeppoSAX and Swift Burst Alert Telescope spectra of the Seyfert 2 galaxy NGC 4945, but the basic methodology is applicable to other AGNs, including Compton-thin sources. The models overcome a major restriction of disc-reflection models, namely the assumption of an infinite column density. Finite column-density models produce a rich variety of spectral shapes and characteristics that cannot be produced by disc-reflection models, even for Compton-thin AGN with column densities in the range ∼1023–1024 cm−2. In the Compton-thick regime, we show that even though NGC 4945 is one of the brightest AGNs above 10 keV, there are significant spectral degeneracies that correspond to very different physical scenarios. The models that fit the data span nearly a factor of 3 in column density (∼2–6 × 1024 cm−2) and 2 orders of magnitude in the intrinsic 2–195 keV luminosity. Models in which the continuum above 10 keV is dominated by the direct (unscattered) continuum give the highest intrinsic luminosities and column densities. Models in which the Compton-scattered continuum dominates the spectrum above 10 keV give the lowest intrinsic luminosities and column densities. Utilizing variability information from other studies of NGC 4945, namely the fact that the Fe Kα emission line does not vary whilst the continuum above 10 keV varies significantly, we can select the solutions in which the direct continuum dominates above 10 keV. The data require that the Compton-scattered continuum and Fe Kα line emission come predominantly from the illuminated surfaces of the X-ray reprocessor, implying a clumpy medium with a global covering factor that is small enough that the Compton-scattered continuum does not dominate the spectrum above 10 keV. The line of sight may be obscured by matter in the same distribution but a separate ring-like structure observed edge-on is not ruled out. The Fe Kα line-emitting region must be the same one recently reported to be spatially resolved by Chandra, so it must be extended on a scale of ∼30 pc or so. As found in previous studies of NGC 4945, the implied intrinsic bolometric luminosity is close to, or greater than, the Eddington luminosity. However, a scenario that is also consistent with the data and the models is that NGC 4945 is a strongly beamed AGN embedded in a shell of Compton-thick (but clumpy) matter, with a covering factor that needs less fine-tuning than the case of an isotropic intrinsic X-ray continuum. The intensity of the intrinsic X-ray continuum would be strongly aligned along or close to the line of sight, so that the true intrinsic luminosity could easily be an order of magnitude less than that deduced for an isotropic X-ray source. Beaming also appears to be consistent with recent radio and Fermi results for NGC 4945. Such beamed Compton-thick AGNs would be preferentially selected in hard X-ray surveys over unbeamed Compton-thick AGNs.