I'll be brief......
Most schools, both public and private, still call their teachers by Mr./Miss/Mrs.(Surname). A few teachers now and then will have a school-wide accepted nickname. (Of course a lot have nicknames not mentioned formally. LOL. In fact, I've come up with a few myself, and that was while I was teaching!)
Not all private schools are equal. It's a fallacy that all private schools are better than any public school. There are some good public schools and there are some horrendous private schools.
Sonic, I know those schools intimately from both sides. I know you innocently spoke (or typed) but you fell for a very common misconception as to what is so wrong with them. Not that all teachers try to or can teach. It's just like any other job. Some can do it and some can't. There is just so much else fucking up the system that most people have no clue is even going on.
Personally, I think we need more apprenticeships that train and lead to real jobs, and a little less college herding that often invests a lot of money and time but leaves the students lost and bewildered. Real training, not mass marketing training. Don't get me wrong, I love learning, I'm just disillusioned with the business of higher learning.