The sense I got from a friend of mine who finished the program is that they do view it as a 'legitimate' masters, so to speak. This is in spite of it being a masters from outside of the United States and it being shorter. I am referring to the MSc masters which is one year, but obviously two years is even stronger. Many friends of mine found work in the US no problem after doing a masters at LSE. A very good friend went to work for the British embassy in DC and is now at Google in Mountain View, CA.
My sense is that it would be very unlikely to run across someone in government administration / non-profit / international organization / foreign service / etc. anywhere in the world who has NOT heard of LSE.
I made sure to do this kind of asking around before applying, as I am an American citizen and could end up back there after my masters. However, I am not yet sure where it is I would like to live following the masters, so having a name on my resume that is recognized across the world is very important to me. This is why LSE and Harvard were my top choices. As Harvard is not offering any money and I have a full ride to LSE, I will very likely end up enrolling at the latter.