Now based on what you have described, we can see drooling from one side associated with a range of issues. Now I am glad to hear that he is swallowing normally and that his tongue is not compromised, as some neurological conditions can cause what you are seeing. With that aside, the most common issues we'd have to consider are secondary to focal injury, trauma, rotten tooth, dental
abscess, something being caught, or irritation (from an insect bite, something he ingested, or something being caught) on that side of the mouth. Otherwise, we can see this with inflammation of the salivary gland on this side.
With all this in mind, it is not an urgent situation but you will be limited with what you can do at home. To rule out irritation from any oral exposures, we'd want to irrigate or flush his mouth out with cool water. While doing so, we need to check between the teeth, under the tongue, and the root of the mouth for anything caught. Anything found needs to be removed.
Otherwise, to address any possible reactions to an insect sting/bite, you can try him with an antihistamine. Most commonly we use Benadryl/Diphenhydramine (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/diphenhydramine-benadryl). A low dose (ie. 0.5mg per pound of their body weight twice daily) can just be enough to reduce allergic irritation. We like to keep the dose low, as it can cause drowsiness (just like people). And of course, this medication shouldn't be used if your wee one has any pre-existing conditions or is on any other medication without speaking to your vet first.
Overall, these would be our concerns for his sudden onset drooling. And if anything is present irritating the mouth on that side, then this will lead to this one sided drooling that we are seeing. Therefore, we can use the above, but if he isn't settling then we'd want to move that appointment up. That way his local vet can assess his mouth and this over-reactive salivary gland. Anything caught can be removed and if dental issues or infections are suspect, then dog safe pain relief +/- antibiotics can be dispensed to tackle that (+/- removal of any bad teeth) to settle this for him.
Please take care,