If you have a pyramid mipmap file texture, the image will be rendered at full resolution and at smaller resolutions as the object moves away from camera or objects in the Bg as Chris explained.
Today's renderers can do this on the fly to save on memory, as an example Arnold uses .tx but there's also exr, tiff and dds . Also remember that having large image textures will slow down Maya opening, saving and viewport performance, after that comes the render engine that needs to load everything into memory.
There are many ways to optimize one's rendering performance depending on the job at hand.
You can also covert manually any texture to a mipmap format as long as they are power of 2 to help the rendering but viewport performance will remain the same because the texture is always at full resolution unless you control it in the viewport 2.0 settings "Clamp texture resolution".
There's no need to use a copy of a 4K texture at a lower resolution for another object since you already loaded the 4K texture into Maya anyway, so just keep using in as many objects as you want within the scene. You only need smaller resolution for distant objects.( but the renderer can also take care of that as explained prior). You can read more about this in the docs or online.
