Private jets nearly always have a dedicated terminal and customs boards to check your passport directly or at their dedicated terminal, so you bypass customs that the masses use. Sometimes, you're automatically cleared at check-in and no-one checks on landing.

I can't think really think of a scenario where they would disclose your name to your client, much like if you were passing through 'regular' customs. Also, your client should respect your privacy enough to move away when your passport is being checked anyway.