The condition we've encountered with our rescued Pyrs is something called Entropion. Chows, Shar-peis, St. Bernards, and Bull Dogs are all predisposed to this. With the hundreds of Pyrs who came through the rescue, I can personally recall about maybe 5 Pyrs with Entropion. What happens, is that the top eyelid (dogs only have eyelashes on the top) curls under and the hairs scratch their eye's surface, thusly the tearing. Untreated, it can scratch the corneal surface to a degree which may cause damaged vision or even blindness if left long enough. Some young pups can be born with this, and end up outgrowing it in a short period of time; it would hardly be noticed.
While usually congenital, it can be acquired through trauma or introduction of a foreign body. This can also result in "pink eye" (conjunctivitis) and other conditions. It can be treated with anti-inflammatories and/or anti-biotics. In it's congenital form, surgery is usually required to fix it. Think of it as an "eye lift". It's not a serious procedure, but it's not cheap. It is, however, necessary to ensure your dog's eye is not damaged. Recovery from the surgery is minimal. Visualizing if the eyelid is curled under is easy, in my experience. You can easily take a close look and determine it. I'd be willing to bet, though, that less severe cases of Entropion may be harder to spot and you'll definitely want to ask your vet if you see tearing.
Conversely, there is the opposite of this condition called Ectropion which impacts the bottom eyelid versus the top, and the lid is everted. Also congenital, this dog may not be able to properly keep their eye moisturized while blinking and can get chronic dry eye. This condition is common in loose-skinned dogs; we think of Bassets, Hounds... that "droopy eyed" look. If not congenital, some dogs can acquire it from fatigue of facial muscles (hunting dogs) or due to paralysis from damaging a facial nerve. Often this condition doesn't need any treatment. In cases where the health of the eye is compromised, surgery may be needed to replace the lid against the eye.
So while sometimes tearing can be normal, there are other things which may be the culprit. Always be concerned about bilateral tearing, and any time the tearing is cloudy or mucousy. Always have your vet check out your dog's eyes to determine if their tearing is normal, to rule out any infections or eye conditions.