The only Part 91 recurrent training and testing requirements are in subpart K, so if they don't apply to you then only the general currency requirements in Part 61 would apply.
The SIC currency requirements in 61.55 are (every 12 months):
(1) Become familiar with the following information for the specific
type aircraft for which second-in-command privileges are requested—
(i) Operational procedures applicable to the powerplant, equipment,
(ii) Performance specifications and limitations.
(iii) Normal, abnormal, and emergency operating procedures.
(iv) Flight manual.
(v) Placards and markings.
(2) Except as provided in paragraph (g) of this section, performed and
logged pilot time in the type of aircraft or in a flight simulator
that represents the type of aircraft for which second-in-command
privileges are requested, which includes—
(i) Three takeoffs and three landings to a full stop as the sole
manipulator of the flight controls;
(ii) Engine-out procedures and maneuvering with an engine out while
executing the duties of pilot in command; and
(iii) Crew resource management training.
Practically speaking, what the FAA requires is only part of the picture because your insurance company will probably mandate certain training and currency requirements that may go well beyond the regulations.