Thou politics exist in both scenarios, the private school scenario usually is better managed due to the fact that private school coaches have more control over who plays or comes out for their teams. In many states, private schools attract the best players. Florida for example, with Westminster Christian, where Alex Rodriguez played has a constant flow of high caliber players transferring in. Other states, the public schools have the better players overall. The knock on private schools is the level of competition from top to bottom on the schedule.
Many schools face inferior competition throughout the year.
As far as whether scouts look at private school players, the answer is yes. I had a pitcher out of Greenbrier Christian Academy in Chesapeake, Va. drafted two years ago. Private schools programs are rated mainly on who the coach is. Many schools will hire ex-big league players to coach. With the name recognition and credibility and starts the flow of interest in your program.