There are only a few places that I go where people recognize me anyway. I have to be at the right place, in the right city. It's not really that much of an issue.
Accusations are made directly to Rome about theologians from persons who are not theologians. Some of these accusations are anonymous. The local bishop should be the one to relate to theologians to determine orthodoxy.
Even people who feel perfectly comfortable investing in the stock market and owning their own homes often have qualms about individual medical accounts or Social Security private accounts.