I would think profs would be flattered to have their words made public. It's no secret; it's public, what they say.
I teach, and I haved great fun making fun of political correctness, questioning global warming with opposing facts and viewpoints, and slammo-dunking the idiocy of communism every chance I get. I would like nothing better than to have someone secretly taping me, so the words could be played to a larger audience (and I would be given the chance to defend them -- in the press, or otherwise).
Oops! "haved" = "have had"
Yes -- they say it in public, they get paid to say it to YOUR kids. But they don't want to be held responsible for what they say. Hmmm, cannot have it both ways. What if a student were recording a lecture for study purposes? Often they do.