Speaking from experience on the farm, during the winter months where our shoes/boots would get soaked from melting snow and needing them dry for the next day: putting them on/near a furnace register works well.
Make sure the air is blowing on or at the shoes. It doesn't even have to be into the shoes, although that works best. Just getting the heat and air movement will help them dry quickly, as in less than 12 hours. Wet shoes at night turn into dry shoes in the morning. Well, unless they were really wet.
You can also use a hair dryer or heat gun on it's lowest setting to help the drying process. Don't leave the hair dryer or heat gun running unattended, since they can start fires. Because it's a manual process with your presence being required, it's not going to fully dry your shoes, unless you plan on spending hours doing this, but it'll help speed the process.
In the summer months, when we worked in the rain (or worked with manure), we usually just wore rubber boots. If that wasn't available, we'd just leave the (clean) wet shoes on the back porch, which was indoors, and the heat of the day (usually) took care of it for us.
Running a fan with the air blowing across the shoes makes them dry quickly. Basically, any air movement and/or heat in or around your shoes is going to work.
I haven't tried this one, but it seems viable. Wrap a dry bath towel around your shoes and put them in the clothes dryer on low heat. You can try using a belt or velcro to keep the towel around the shoes. The towel is just there to help with the noise in the dryer. I think some dryers even have a shoe setting or basket.