Edit: neighbor is not home...
Edit again: I flooded the fire pit with a copius amount of water. The pit was still smoking, so I put more water on it until it's be sure that it was out. I went back and checked about ten minutes later and there was still a bit of smoke.
Then I realized that their fire pit was near the base of our tree, the trees roots are huge and some are near the surface. The fire pit is directly over one of these roots that is near the surface. With even more water added I think I got it completely out now, but will continue to check.
I'm afraid that the smoke may have been from the roots burning!