I would say that regarding the criteria you are asking LSE and LBS, also in terms of international prestige, are surely above Duke MMS.
Especially if we talk about alumni network and global reach.
Now, LBS, being a business school has a more "practical" approach compared to the the LSE MiM who has been shaped for people coming from different backgrounds (enginering etc....) . Of course more theoric educations does not necessarily mean less practical, since you have to keep in mind that LSE being a 2 years long course (while LBS is 1) has mandatory internships to do and a very well developed career development service for the students of that course. I think that LSE offers a more in-dept preparation if we talks about a MiM (of course if you are looking for something very practical than consider an MBA). Furthermore, LSE MiM for the opportunities it gives after graduation and also for the possibility of the CEMS double degree is regarded as more prestigious compared to the LBS, who on the other hand is more appreciated internationally for its MBA.
For the career prospects you have both LSE and LBS offers great possibilities in any markets, and the biggest companies are actually looking there among the recent graduates to hire at their associate or analyst level. In my opinion, LSE right now has better links if you are aming at that particular career. So to sum up I would go for it.
LSE MiM and LBS MiM; are both regarded as a "Little MBA" cos are a sort of "accelerating tool" for your career if you are a young and promising student, recently came out of undergraduate or with relatively low work experience, since in order to do an MBA you need several requirements (first of all a discrete numbers of work exp).
Many consider a MiM a substitute, (the "European version") of an MBA, but I would say that it is not like that. doing a MiM does not preclude you from doing an MBA or a PhD later of course, that I advice you to do, if once you will have started working are aming at climbing the executive ladder.
From my exp and the people I know coming from both uni, i would finally say that LSE is better for MiM in terms of theoretical and technical knowledge, and links with the consulting environment, while LBS is better for MBA. My advice is that if are young and not experienced person, and you want to accelerate your career now then go for the two year LSE MiM, that prepares you better, otherwise work for few years and consider doing an MBA at LBS.
hope it clarifies your doubts