Have you noticed that cosmopolitan is of Greek derivation, cosmos and polithis? It can mean international, but generally speaking it means, at home in many places, not limited to the viewpoint of one country and very often it has the connotation of sophisticated. I think you would need to be rather pretentious to use the phrase " We are cosmopolitan". If an international student said " I am a cosmopolitan student" he might run the danger of being mocked.
There are two posts by jeanprendiville on this page which give a good distinction between "international and cosmopolitan". As well as a good example of the use of "διεθνείς" ("international"). And to expand the idea of "cosmopolitan" being Greek, and of course, it is, it's "κοσμοπολίτης". (Citizen of the world literally but more usual "at ease in any part of the world" etc.)