That is to say, is it a 2D thing that always faces the camera? Or is it a 3D thing that might have a better side for viewing, and would need to be rotated correctly to face the camera if the camera never changes perspective in 3D space?
Yes they are billboards:
The Line Renderer does not render one pixel thin lines. It renders billboard lines that have width and can be textured. It uses the same algorithm for line rendering as the Trail Renderer.
Taken from: http://unity3d.com/support/documentation/Components/class-LineRenderer.html