Are private lessons necessary for a student to succeed?

Private lessons certainly make success a lot easier to achieve. I teach at a couple California State Universities. Most of my students are education majors and will not be faced with the postgraduate marketplace competition that performance majors will be up against. I'm certain that the situation is very different at a major conservatory!
 
For some incoming freshman, I am the first one-on-one, private teacher they have seen. Those who have had good private instruction are almost always "ahead of the game." Some of the "first-timers" catch up by graduation – some do not. The ones that do usually have to work very, very hard to do so.  A music degree is enough work as it is – why make it harder!  Also, over the years, I have seen a few exceptions who have put things together pretty well by themselves – talent and hard work will assert themselves.
 
​Find the best teacher you can. A college music major can be a good choice for a young trombonist.