We present radiometrically derived V-band geometric albedos and effective radii for 32 Jovian Trojan asteroids, using near-simultaneous mid-infrared and visible observations. We sampled the large end of the group's size distribution, down to a radius of 25 km, using 14 objects in the L4 swarm and 18 in the L5 swarm. We find that the albedo distribution is much narrower than previously derived from IRAS measurements. The Trojans, for the most part, have very similar albedos. The actual mean and standard deviation of the distribution depend on the average Trojan beaming parameter η. The ``standard'' value of 0.756, which was used for the IRAS analysis, yields a mean albedo of 0.056+/-0.003 and a standard deviation of 0.009. However, a value of η=0.94, which we found represented our data better, yields 0.041+/-0.002 and a standard deviation of just 0.007. The thermal behavior of the Trojans seems to follow the ``slow rotator'' model, and the thermal inertia itself can be no greater than about half the Moon's value. The Kolmogorov-Smirnov test was used to compare the Trojans' albedo distribution with that of cometary nuclei, dead-comet candidates, and outer solar system objects. We find that the Trojan distribution is similar only to the cometary ones, and only if the Trojans' η~1. Observations of the binary (617) Patroclus reveal that its albedo is rather typical among the distribution. We have also discovered that (4709) Ennomos has an extremely elevated albedo, about 0.15. This object may have a very unusual thermal behavior or have recently suffered a large impact that excavated the surface down to a layer of highly reflective, pristine ice.