Chickenhouse - I agree that while no sane person operates a decent airframe without hull and liability, it is possible for me to name and shame certain people who do so under part 91 rules - one with a King Air and one with a Lear. This is for people who do not finance the aircraft. No sane bank would permit you to fly an asset without it.
Under FAA rules, it is not a requirement. Therefore should the aircraft remain in FAA airspace, it doesn't need one. But that is provided they operate under part 91 only.
Should they be operating for hire - 14 CFR 205 states they should have liability insurance and that a copy is filed with the FAA, and a copy is maintained at the operators office.
That being said, FBO's want proof of insurance when you are using their facilities, as do other aviation services.
That said, given we operated globally, we always had hull and liability insurance, and a copy of the policy was onboard. If necessary, a translated copy was also onboard in the aircraft folder. (I think Italy required a translation).
It does vary from authority to authority around the globe.