Hello, I'm a student attending Bocconi and I think the university is definitely more known in Europe than in the US
Nonetheless, some of the best US universities (among which Columbia, Yale,Princeton and so on) are partners in the Exchange Program of Bocconi...so I think it has a pretty good reputation overseas too.
In my experience, professors and TAs are well-prepared and always willing to help and clarify your doubts.
Lessons are interesting and stimulating - with some exceptions of course - and extracurricular activities are offered as well (sports, foreign languages, seminars, IT courses and so on).
The structure in itself is well-maintained, not overcrowded nor damaged at all. On the contrary, it's quite well-organized (unlike most italian universities).
I'd say the key strength of Bocconi is the placement, since it has lots of contacts with national and international companies..lots of company presentations and career events (where you can meet employers and drop your CV) are arranged.
Of course it's not all a bed of roses. Milan is not such an international city as it pretends to be, some parts of the city are really dirty and chaotic....and many Italians speak english but very few of them speak it properly. Finally, it's true that Italians account for the largest part of the student body...but BIEMF is not the only english taught course (the majority of the master's degrees are taught in English), so you'll find a goodly number of international students around!
I apologize in advance for any spelling mistakes eheh
Hope this will help you in making your choice!