How does security and baggage checking work in regards to a private jet?
Here in the US (you didn't specify where, and this is different in other countries), traveling domestically in your own private aircraft is much like traveling in your own private car.

Except in very specific situations, you are not generally required to go through security, and at many, many smaller general aviation airports airport security is almost non-existent, or at the very least they are not checking passengers and their luggage.

If the police have a reason to suspect that something illegal is taking place (they have reasonable cause) then they can search the airplane, but again, much like in your car, this doesn't usually happen.

International flights are a very different matter though, and the airplane, crew, passengers, and luggage must all clear customs at departure and arrival airports where they will be inspected.