How does security and baggage checking work in regards to a private jet? I was told recently that if you are flying on your own private jet, you can drive your car directly on to the runway, park next to your plane and board without any security checks.

This can't be true can it? 

I don't own my own private jet so I'm afraid I can't check for myself.

How does security work around privately owned planes landing in public airports?
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.