I think that you can teach yourself a subject quite well from the internet, videos, etc. I do not believe that you can reach mastery in a subject though. This is because there are a couple of things missing. First off, I believe that by teaching yourself and not having interactions with professors you might miss patterns that develop in a subject, or other aspects that are more intuitive than anything else. Second, you won't receive criticism on your studies that might help you look at a problem in a different light, or change your routine for solving a problem to a more efficient one.
I agree with you on the first point, but I think that what you describe in your second point can still be found online. If one were to post in a forum relevant to the subject being studied, it is likely that you will receive responses that show you a different method of solving a problem. The people responding will most likely have received some type of education in the subject, so the information that they are passing along is quality information, albeit second hand information.
I do suppose you are right that anyone can claim they are an expert online. However, in certain areas such as math or (some) sciences, this claim is verifiable by seeing if what they said works out! Sometimes it just takes someone pointing something out to make things click, and in reality that person need not be an expert.