Whenever I discuss such matter with Americans, I conclude that comparing LSE with Oxbridge is like comparing MIT with Harvard + Yale. As with MIT, which focuses on sciences (although it does have an excellent management program), LSE is a specialized social science school. But, Oxbridge, Harvard, and Yale are large universities with many more subjects being taught. So, it's difficult to compare the two types, unless one focuses solely on courses like economics. In that case, LSE is regarded by educated Americans (including current Ivy Leaguers) to be THE place to study economics and other related fields. Many of my friends, who will be attending top Ivy League schools next year, are planning to spend a semester (through an exchange program, I presume) or at least pursue a master's degree at LSE.
But anyway, to answer your question, for the people in general, Americans do not know what LSE is. However, for somewhat to highly educated individuals, they all know what the school is and regard it with utmost respect. For them, hearing people say, "Oh, I will be attending Yale next year" is a trite these days, but when I say, "Yes, I will be pursuing a degree at the London School of Economics," they think it's VERY prestigious and exclusive (probably because not many people go to LSE from the US). For example, I was at a fund raising cocktail party last week, where I had discussions with many people who were working at companies like Goldman Sachs, Lehmann, JP Morgan, Deutche Bank, etc, they semi-bragged about their occupation then asked me, "Where will you be going to school next year?" I said, "I will be attending the London School of Economics to pursue a degree in finance." All the 35+ year old, rich i-bankers agreed that it was "the best school" and some even gave me their business cards. My friend, who will be going to Yale and was standing right next to me, did not receive any praise other than "that's good for you" and certainly did not get any business cards (even though he's planning to study economics). So, in that regard, if you talk to people who are in the pertinent area of interest, LSE is even better than Oxbridge and the Ivy League Schools. Perhaps when British people meet someone from Harvard, they'll think it's better than LSE. Well, it's vice versa in the US. It's all about the rarity these days.
Anyhow, in conclusion, normal people (random girls I meet at a party) do not know what LSE is. However, the ones who matter do perceive the school as one of the most exquisite places to study social sciences. Again, it's like how people regard engineering majors at MIT to be "geniuses" while they think engineering majors from Yale to be somewhat lacking.