One reason why people ask questions in private is for fear of looking incapable and dumb. It's not an entirely unfounded fear when you consider that we're knowledge workers. Our worth and opportunities are heavily influenced by how others perceive us.
Most people are probably aware of the benefits that others would get from asking questions publicly. The people that are not aware may pick it up when you publicly post the original question and answer yourself, e.g., "Someone recently asked me how to make a perfect soft boiled egg. Here's how I go about it."
I always had this policy when I was teaching college classes.
Obviously, the rule didn't apply to personal problems (grade worries, student (or family) illness), but if I got a generic question about an assignment in email, I'd ask the student to repost it in the online forum for the class.
It worked very well -- it's a rare assignment question that only one student is wondering about (whether they actually ask or not). It cut down on the number of emails I had to answer, saved time for the other students (they were supposed to check the forum before asking questions), and gave me a central location to check when I was prepping/improving the course for the next semester (rather than being buried in the Satanic depths of my email box, all of the questions and clarifications for that particular course were collected in one spot).