I downloaded the project and found the issue: In jungle_dirt_tileset the StaticBody2D and CollisionShape nodes are scaled, which is causing the problem. It seems the CollisionShape2D scale isn't causing an issue (surprisingly), however the StaticBody2D scale is being reset to a scale of 1 when used with the Tileset, which is leading to the large collision shape margins you are seeing.
The fix is thankfully simple: Just set the scale of the StaticBody2D nodes back to 1 and adjust the collision shape nodes as needed. Personally, I'd reset the CollisionShape2D node scales and just change the Extents property of the RectangleShape to X:16 Y:16 just to be on the safe side.
With those fixes, the collision should work as expected:

Edit: I forgot to mention I was using Godot 3.1.1, which may have helped. The Tile set scene still needed adjusting, but updating may have helped.