If a pilot has the opportunity to fly as an SIC in a private jet, what FAA requirements must they comply with in order to fly with passengers?
Theoretically you can fly a Boeing 747 on a Private Pilots Licence as long as you have the relevant type ratings. If you are flying passengers for profit you will then require at least a Commercial Pilots Licence.
Modern day training courses which prepare you to be the first officer or co-pilot as it were, tend to result in you holding a frozen ATPL, ie, you have done all the exams and you're now hour building in order to qualify for the licence. You will also likely have completed a Multi-Crew Co-operation course of some description, obtained your jet rating and hold any relevant Instrument Ratings. It probably depends on which provider you were choosing to apply for.
EDIT: Ah, my apologies, this post was in relation to European (EASA) guidelines (more specifically in the UK really).
In general, FAR Part 61, section 55a,b outlines specific requirements for those designated as Second-in-command of Aircraft requiring more than one pilot. Section b probably covers the specific training you're curious about. See also parts 121, 125, and 135 for more specific requirements.