It is the tax rules that apply in the place where you are tax resident that determine the information you must provide to the tax authorities and how you must provide this information. You will normally be required to declare information concerning your financial accounts in an annual tax return. If you are a US citizen, you are obliged under US tax rules to submit a tax return to the US tax authorities (IRS) every year.
You cannot avoid your duty of declaration abroad, even if information concerning your financial accounts in Norway is sent to the tax authorities in your country of residence (and the USA if you are a US citizen). You must contact the tax authorities in the country concerned if you are in any doubt as regards whether you must declare information to the tax authorities in that country.
Taxation may be regulated through a tax treaty if you have income and assets in several countries. Tax treaties that Norway has entered into with other countries can be found here.