“Generally speaking, private school teachers are more motivated and better trained to write more effective LoRs”
I specifically asked this question at my kid’s GLADCHEMMS private school. Per the school there is no teacher training and no tracking of outcomes to motivate effective LoRs. Other schools may do this but I doubt they do…
However, the class size and the number of students asking for recommendations will definitely affect their quality. Many private schools have smaller classes, teachers as dorm faculty, teachers as coaches, etc. This allows the teachers to get to know their students more holistically, which may help strengthen the recommendation. In addition, a teacher writing 10 recommendations will likely do a better job than a teacher asked to write 40.