Hey there,
I am currently in my first year at Hertie and am very happy with my decision to come here (I also had an offer from LSE). Not having been at any of those other school you mention, it is hard for me to tell you how we compare. What I can tell you, is that I know that several of our dual degree students that we have here are very happy with the school. One of the students that are here from Columbia/SIPA said he felt he was getting a better education here than back in NYC, and also the other SIPA- students seem generally very satisfied (We also have a couple of dual degree students from LSE here, but I haven't talked to them that much). We also had an exchange student from Duke/Sanford here last semester, and she said that in her opinion Hertie et least meassured up to that school.
As far as reputation goes, Hertie is obviously not as well known internationally as for instance LSE, the school is after all only five years old. However, this is something the administration is working very hard on, and in my opinion it is pretty impressive how much of a repuation the school has managed to build up in only five years.
Hope this helps you at least a little bit, and if you should have more questions concerning Hertie, do not hesitate to ask!
Good luck wih your choice!