BwanaBob, that is in fact one key, and probably the biggest single one. But there are so many potential noise sources for an instrument like this, that even if you see something at both sites at once, the most likely explanation is that they both just happened to have some random noise at the same time.
You could do multiple passes this way with even more instruments, except that there are currently only two LIGO sites. There are a few similar instruments in other countries, and they do look at the data from those, too, but none of them are as good as LIGO, so it doesn’t really help much. There have been proposals to build a third LIGO in some other country (most likely Australia), and that would probably be the best bang for the buck for improving the data (a third one wouldn’t cost nearly as much as the first two, because all of the R&D has already been done), but the international aspect makes it politically difficult: The NSF is reluctant to spend money outside of the US, and while the host country could also contribute some share, that then means that you’ve got two different governments you need to beg from.