Last night I was speaking with a kind gentleman who, as he spoke, collected copious amounts of spittle in the corners of his mouth.
I know three other people who have this “issue”, and no matter how hard I try to ignore it, I can’t.
Is there a medical term for this? What causes this? The only common trait I have noticed between all four people is how they speak- their mouths don’t seem to shape like other non-spittle collecting people.
There is a medical term for everything.
I too, have encountered this in two individuals, it’s kind of discomforting.
(waiting for answer)
If the problem is due to excess production of saliva, then ptyalism is the correct term.
If it is due to improper formation of the lips, then I propose St. Bernardism. 
BTW, the absolute lack of salive (dry mouth) is called xerostomia. I always have had a desire to kiss someone with this condition. Just for kicks. 
I suspect that’s because you’ve never seen someone with xerostomia. It’s nasty! In extremis, one sees eroded, pitted, ulcerated bleeding mucous membranes, adhering to other structures in the mouth, with the fluid present being derivatives of white cells and ultrafiltrates of blood mixed with bacterial detritus (aka pus).
Fortunately, artificial saliva can help a lot. Sadly, the inability to dose with artificial saliva during sleep makes full recovery more difficult.