With our current possibilities going near speed of light is just as impossible as going faster than it, so why not assume the more practical way, seeing as both are equally impossible right now.
The way I see it, where there is actual matter, it 'displaces', for lack of a better word, the empty space that was previously there. This means that, while the empty space expands, it may push objects further apart, but it doesn't stretch them out, so distances on earth would stay the same. Of course I am no physicist, but this is how I think it works. I imagine it like balls floating in a pool of water. If you pour the water in a bigger pool it consume a wider area, but the balls stay the same size.