I don't know what most people use on the mats, but in the tattoo shop we have to use products on our surfaces that list "germicidal, fungicidal, bactericidal and virucidal" and most importantly "Tuberculocidal " that usually covers all the previous ones. The product still has to sit on the surface for 5-10 minutes to be effective, then can be cleaned off. A Bleach solution is also effective, but im not sure what it would do to the mats over time.
Either way i like to shower before class, and again as soon as i get home, thats about all i can do on my end.