Is it legal for a private pilot to advertise that (s)he offers free passenger flights?
"Holding out" is relevant only in determining whether a flight is private carriage or common carriage, as described in AC 12-120. The definition of common carriage used by the FAA is:
- a holding out of a willingness to
- transport persons or property
- from place to place
- for compensation
If you receive any compensation for these flights, you would fall under the definition of common carriage and this would be illegal. The FAA interprets "compensation" very broadly, including:
- Reimbursement of expenses (e.g. fuel, oil, transportation, airport expenditures, aircraft rental fees, lodging, costs of ownership etc.)
- A free meal
- Logging of flight time when the pilot does not have to pay for the costs of operating the aircraft
- Salary or wages
- Goodwill in the form of expected future economic benefit
So you would need to be sure that you are not receiving any compensation. The passenger can't give you a tip or buy you dinner. They can't pay pro rata share of the expenses. If you have any business relations with your passengers in the future, you can expect them to fall under a great deal of scrutiny.
As long as you can avoid being compensated in any sense, this should be legal. Although I would be wary of doing so anyway, since the thought of having to fight the FAA in court over this should certainly give you pause. If this is something you are actually planning on doing, as opposed to just a thought experiment, it may be wise to consult with a lawyer before starting.