Private schools offer a variety of different benefits, too, and each school or company come with their own pros and cons. I work at a school that focuses on technology, so I was given a tablet to use in the classroom that had all of our lessons already installed. Finding the ideal school is really up to the individual. Consider yourself a free agent that is free to sign with any team you want. You can go straight for the most money, or perhaps a beautiful location like Busan or Jeju Island. I have heard from teachers I have met at other schools – as well as from interviews and job postings from before I started at my current school – that some other private schools offer vacation time, professional development leave, training, rent-free apartments, airfare reimbursements, and shorter work hours. Many private schools do work later hours since the students don't arrive until their public schools are dismissed, but that really has not been an issue for me because the vast majority of places in South Korea are open extremely late. In fact, my unique work schedule enables me to avoid the big crowds just about everywhere I go on weekdays.
Additionally, teaching in a private school offers many opportunities to meet new people. The school that I teach at has 14 foreign teachers, and each has been a tremendous help as I get acclimated to life in a new country. My coworkers have shown me their favorite bars and restaurants, hikes in the nearby mountains, and have even helped me learn Korean – although I'm definitely still a beginner. When teachers leave, many of them give away or sell the belongings that they cannot take with them, so I have gained couches, TVs, and even a PlayStation from my departing coworkers. My school also has Korean staff members who interact with the parents and handle most of the important paperwork, which makes is much easier for me as a novice Korean speaker.
Even if you are still wary about private schools, take an interview (or two). For some people it can be an ideal opportunity, and there's no harm in seeing what's available.