For private aviation the responsibility for security of the flight falls on the operator of the flight, which usually means the crew is handling any "screening" of passengers. Typically if you're flying on a private aircraft you're a known quantity: Executives flying around in GM's jets are going to be GM employees (or their guests), folks flying around on a sports team's jet will probably be the players and coaches, etc.
In both of those cases chances are the passengers have met the pilots before, and neither they nor their guests are likely to try to hijack the flight so there's no need for TSA-style searches and confiscation of the bottle of 30-year-old Glenlivet they brought to share on the flight. Similarly for charter flights the folks renting the plane will have spoken to the company extensively in arranging the flight. There may be some additional level of scrutiny here to make sure someone's not bringing a propane camp stove or something else prohibited on board, but chances are you will not be asked to remove your shoes and belt. The level of security for the airport itself will vary depending on the airport (and this is what usually determines if you can drive right up to the aircraft).
Let's consider three examples (for the sake of a prison metaphor we'll call them maximum security, medium security, and minimum security): A maximum security airport would be the "big" airports (like JFK or O'Hare) - At these airports chances are you will not be driving your car or limo onto the ramp to board your plane. You will park (or be dropped off) "landside" (outside the airport fence / operations area), and authorized persons will take your bags and escort you to and from the aircraft.
When not on the way to or from the aircraft you'll be in a nice terminal building. The medium security airports are typically the larger general aviation fields and/or the smaller commercial/airline fields. If your aircraft is based at one of these airports you will typically have reasonably unfettered access to it (subject to a few restrictions, such as the need for a background check and ID badge). You will usually be able to drive out onto the ramp and go directly into the aircraft. If you're not based at the field (for example on the return leg of a business trip) you may be allowed to drive out on the ramp with an escort from someone at the airport who is authorized to do so, but visiting crew and passengers will be under some level of supervision from airport personnel (either directly escorted or visual observation).
Minimum security airports are more associated with light general aviation than business jets, though you'll find private jets based at some of these fields. Security is generally much more relaxed at these airports, and at most of them you simply sign in and drive through a gate right onto the ramp. Good common sense says passengers will usually be escorted by some qualified personnel at these fields (after all you don't want them walking into a propeller or getting sucked into a jet intake), but the supervision is probably going to be provided by your flight crew rather than airport personnel.
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