The FAA is only authorized by congress to regulate FLIGHT, not radio communications.
Radio communications are regulated by the FCC.
Both of these agencies have no power to make laws. They can make only "regulations". In the United States, according to the Constitution, laws can only be made by Congress or the legislatures of the several states.
The position of the FCC is that it is legal for any person to monitor and record any radio transmission that they wish. The only exception to this is listening to private telephone conversations or encrypted military transmissions. The Federal law that prohibits this, explicitly exempts maritime and aeronautical transmissions. This is the language:
18 USC § 2511 (2)(g.ii.IV) It shall not be unlawful under this chapter
or chapter 121 of this title for any person to intercept any radio
communication which is transmitted by any marine or aeronautical
Therefore, by United States federal law it is explicitly allowed and permitted for anyone to monitor and record maritime or aeronautical communications.