Context. Determining whether asteroids in cometary orbits (ACOs) are dormant or extinct comets is relevant for understanding the end-states of comets and the sizes of the comet population. Aims: We intend to study the value distributions of effective diameter (D), beaming parameter (η), and visible geometric albedo (pV) of ACO populations, which can be derived from NASA's Wide-field Infrared Explorer (WISE) observations, and we aim to compare these with the same, independently determined properties of the comets. Methods: The near-Earth asteroid thermal model (NEATM) is used with WISE data and the absolute magnitude (H) of the ACOs to compute the D, pV and η. Results: We obtained D and pV for 49 ACOs in Jupiter family cometary orbits (JF-ACOs) and 16 ACOs in Halley-type cometary orbits (Damocloids). We also obtained the infrared beaming parameter η for 45 of them. All but three JF-ACOs (95% of the sample) present a low albedo compatible with a cometary origin. The pV and η distributions of both ACO populations are very similar. For the entire sample of ACOs, the mean geometric albedo is p̅V = 0.05±0.02, (p̅V = 0.05±0.01 and p̅V = 0.05±0.02 for JF-ACOs and for Damocloids, respectively) compatible with a narrow albedo distribution similar to that of the Jupiter family comets (JFCs), with a p̅V ~ 0.04. The mean beaming parameter is η̅ = 1.0±0.2. We find no correlations between D, pV, or η. We also compare the cumulative size distribution (CSD) of ACOs, Centaurs, and JFCs. Although the Centaur sample contains larger objects, the linear parts in their log-log plot of the CSDs presents a similar cumulative exponent (β = 1.85 ± 0.30 and 1.76 ± 0.35, respectively). The CSD for Damocloids presents a much shallower exponent β = 0.89 ± 0.17. Conclusions: The pV- and η-value distributions of ACOs are very similar to those of JF comet (JFCs) nuclei. The ACOs in Tancredi's list are the best possible candidates to be dormant/inactive comets. The CSD for JF-ACOs is shallower and shifted towards larger diameters with respect to the CSD of active JFCs, which suggests that the mantling process has a size dependency whereby large comets tend to reach an inactive stage faster than small comets. Finally, the population of JF-ACOs is comparable in number with the population of JFCs, although there are more tens-km JF-ACOs than JFCs.