Left: Using global space. This is the default behavior; it's what happens when the local button is not activated. Right: Using local space, with local button activated.
In these last two images, the constrained bone (shown in green) is actually a child of the root bone (the bone at the beginning of the chain). This demo shows possible uses for the location constraint in a rig. Note that the green bone still inherits rotation from the root because the root is its parent. This is by design though; the green bone could be the child of any of these bones, or none of them.