From the comments: "... how much noise exists in the measurement environment? I find it difficult to believe something even as mundane as a minuscule earthquake or similar ground noise wouldn’t cause orders of magnitude more of a signal and therefore swamp out any meaning information" This is also the first thing that came to my mind. How do the filter noise? Or are we missing something that is not described in the a…
As I understand it, this is one of the main reasons that there are two detectors thousands of miles apart, in Washington and Louisiana. There will be a whole lot of false alarms at each one separately, but it's extremely unlikely that precisely matching earthquakes will occur at both locations at once. (They have other very sophisticated ways of isolating the system from outside noise, too, of course. But I don't think the experiment would be viable without the two location filter.)
As for space, the LISA project has been trying to get funding for years. (The preliminary "LISA Pathfinder" mission launched on Dec. 3, so there's finally some real progress on this: http://sci.esa.int/lisa-pathfinder/)