First of all, I don't know anything about the MPA program, and none of my friends are doing it. Sorry.
Recruitment into EU institutions as well as world institutions is strong. I hope you are an EU national though, because that's a requirement for quite a lot of the EU positions.
Average income for LSE grads in private sector work is much higher than for those going public sector. If you go to the LSE Careers Service office you can look at salary stats for the last 5 years or so, disaggregated into industry, firm/organisation name, job title, and LSE program completed. The process of getting a private sector job is also quicker, and arguably easier. Anecdote: it is common for LSE students going private sector to have multiple job offers before they graduate. It is fairly common to start interviewing for private sector jobs in September/October (when the MSc program hasn't even started yet, or is just barely underway) and receive one or more job offers before the Christmas break. Many also attend interviews and receive offers between February and April. For public sector job-searchers, the hunt can go on for months after graduation. One guy I know didn't actually get to sign with this government organisation until 13 months after graduation. It's not because he was a poor candidate, that's just how long it takes some public sector institutions to hire a new employee.
Development jobs with global institutions or well-known NGOs, are in my experience around LSE more competitive to get than front office investment banking jobs at the top banks. Development positions frequently require a period of unpaid internships (up to 1 or 2 years) and a total of 3 years relevant work experience before a candidate will even be considered for a paid position. Don't ask me how people afford to live in NYC or Geneva while working for the UN on $0 salary, because I have no idea.
The LSE careers service likes to emphasize that recruitment at LSE is strong across all sectors, which is true. But recruitment is exceptionally strong for finance and consulting positions (arguably among the strongest in the world) while recruitment for other industries is still strong but not on the same level.
Disclaimer: I will be going private sector. However I come from a program in which roughly half or more of my classmates are aiming for public sector.