Carrots are a great snack for labs! When Stella was a pup-pup we would put half a large carrot in the freezer for a bit and then let her chew on it for teething. Now being 18 months, I have cut down her carrot treat to a couple times a week.
Although they are great for her stools, I found that they would sometimes go "through" her so I started to limit the amount given.
We now give her an apple (sliced in half with no seeds or core) and even sprinkle some cinnamon on it (it's her crate snack when we leave or after a good run or swim) - she LOVES it! The Vet said that both apples and carrots are great for their teeth too!
I think it depends upon the dog. Callie digests them, Angus poops them out and King won't touch them.
Our Yellow Baby Girl "Dream A Little Dream" born December 16 2014
Our Chocolate Girl "Kona of the Storm" born August 8 2014
Our Black Boy "Angus Demetrius" born April 26 2013
Our Yellow Girl "Calliope" born January 6, 2006
Our TriPawd "King" - Shelter Rescue born late fall 2004