LBS MFA vs LSE MSc Finance & Private Equity

Both are great schools and are considered to be on par with each other in Europe, so congrats on your offers. I was in a similar position myself.

From a careers point of view, I think the opportunities will really be quite similar - a quick Linkedin search will show alumni from both schools in most places.

LSE is notoriously more academic, LBS more career and networking oriented. That isn't to say the quality of teaching at LBS would be inferior, just more practical. Have a look at the profiles of professors in both faculties to aid your final decision.

Something you might want to consider is the opportunity to get involved in clubs and extra-curricular activities, and the advantages of having a real campus. If this is important to you, then I would choose LBS. If you prefer to get very technical, or you have a specific field of interest you would like to explore further and write a dissertation on (you wouldn't have the option to write a diss at LBS), then I would pick LSE.

The most important piece of advice I can give you is to speak to current students, that helped me make my decision.

TL;DR: It really depends on what you plan to get out of your master's degree, but you can't go wrong with either choice, especially given you already have an EB summer internship secured.