I went to both flavours.
I started at Private which I believe gave the best possible start to my education and instilled in me all those ethical values which a nun-father-chapel institution holds dear.
It showed me a cloistered environment, associates with fabulous wealth and influence, how the old boy network runs, taught me manners, how to be a gentleman and probably affected something resembling good breeding.
I finished at a city State institution. This offered me an insight into an enormously more varied melange of cultures, ideals, goals and backgrounds. It taught me how to become self-motivated enough to survive in the Real World and make money, how to interact in socially disparate groups and how to value contemporaries for their contribution to my life, not to funding a new school wing. It also furnished me with the requisite qualifications to apply to university.
This "foot in both camps" approach has given me a post-education view of the world quite unique amongst my peers; offering the advantages and drive of the satired Silver-Spoon start but incorporating the necessary real-world skills of how to achieve those goals fostered by my start in life.
Doing both is the way to go.