The Ivies' undergraduate programs, in general, are certainly overrated (there are exceptions, like Princeton). In my personal case, being accepted to LSE has been looked highly upon by classmates, faculty, and other adults. Yes, I currently study in the states at a very competitive high school, and people usually consider LSE to be a lot more prestigious than Harvard, Yale, or Princeton. One reason could be that so many students go there each year, and I was the 2nd person in the past decade to be accepted at LSE. However, my point is that if someone in the states recognizes LSE, he/she will most certainly consider the institution to be on par with HYP. For instance, because of me, those who are going to Cornell, MIT, and Penn Wharton from my school have been overshadowed by my "success."
After all, however, it will depend on the person you are talking to. Generally though, LSE is regarded very highly by knowledgeable Americans.