By the mid-1970s we were being spoiled by lighter shoes with nylon uppers that didn't soak up nearly as much water and would actually dry out if you gave them half a chance. You do have to give them that chance, though. In the bad old days, you might as well throw your wet leather training shoes in the bottom of your gym locker and let the moulder, because they weren't going to dry out, anyway. So how do you get training shoes with nylon uppers dry before your next workout? Because you do want them dry, and as soon as possible, if for no other reason than that the current crop of training shoes all smell like cat urine if left if they're run in hard and put up wet.
There are a few methods that you should under no circumstances try, because just about all these shoes are put together with heat-activated glue. So when your grandmother suggests putting your wet athletic shoes on the radiator, thank her for her advice but under no circumstances follow it--and don't leave your wet shoes laying around so that she can helpfully place them on the radiator, on the water heater, in the oven, or in the clothes dryer. The heat from these sources will reactivate the glue holding your shoes together and it will cease holding your shoes together. Another place not to attempt to dry your shoes is in your car under the window. I rediscover this every ten years or so, most recently with a pair of racing flats.
I've been advised that carefully towel-drying the inside of the shoe and then stuffing it with newspaper is effective. It also seems to be a bit labor intensive to me. I'm not one of those people who pre-washes all the pots, pans, and plates before loading them into the dishwasher. After all, if a machine is billed as a dish washer, then it should be able to wash dishes. It's enough for me to remove some of the larger bones before dropping the dish in the machine, right? So I'm not much for the stuffing-with-newspaper method of drying shoes, especially when the shoes came off my feet after a long run. But dry, moving air seems to be enough to dry the shoes. It doesn't have to be desert-dry, nor does it have to be in a wind tunnel. Sitting in an air-conditioned room under a ceiling fan will work. Of course, ceiling fans tend to be in the center of a room, which is probably not where you want your wet shoes displayed (remember the cat-urine odor), especially if you have them on top of a bar stool to be closer to the source of moving air. Refrigerator used to have prominent warm-air exhausts, and a wet pair of shoes placed in front of the exhaust would dry in a few hours. This would probably still work if you could figure out where they are hiding the warm-air exhaust on refrigerators nowadays.
And then there are shoe dryers.
I've been told that these don't work very well. The one that I've been using works great, and you could use it even if you were doing two-a-day workouts in a rainy place like Washington using the same shoes all the time. Which you really shouldn't do because you should rotate your shoes, right? But you could, because that's how well the dryer works. I don't know where it came from or how much it cost because it's quite a few years old, which speaks well for its durability. It was never beautiful, and age hasn't improved its looks, but it lives contentedly in its corner of the garage. It's a plug-in unit containing a small heater and fan, from which two hollow plastic stalks rise. Your shoes go on the top of the stalks, which carry a gentle puff of warm air into the shoes. After a few hours the shoes are dry. There's no noise, so you might forget to turn the dryer off (which you do by pulling the plug; there's no switch). One shortcoming of the unit is that it only dries one pair of shoes at a time. If you're in a household that produces multiple pairs of wet shoes then you're going to need more than one shoe dryer.
An embossed label on the base of the unit proclaims, "PEET The Shoe Dryer." I don't know if they're still around. I kind of hope they are. After all, this unit can't last forever.
Of course, amazon.com still sells PEET Shoe Dryers.