While dogs have a small amount of sweat glands (which are prominently in the paw pads), their primary source of heat exchange (i.e., getting rid of heat) is by panting. Vasodilation (i.e., dilating of blood vessels [which can cause a flushing appearance on the skin]) is another method. Lastly, they are capable of sweating a little via their paw pads.

Panting is the primary method, while vasodilation is likely the second most important. Vasodilation helps bring hot blood directly to the surface of the skin, allowing for the blood to cool before returning back to the heart.

As for heat release and sweating through the paw pads — as a veterinarian — I think this is relatively rare and uncommon. I work with a lot of fit, athletic dogs (e.g., Greyhounds and sled dogs) and have yet to see a dog’s feet sweat much while exercising. So, yes, while your dog has some sweat glands there, it’s likely a minor method of heat release.