Central government:

Norway
Sweden
Finland
Russia

Embassies and Consulate Generals: 

Norwegian Embassy in Stockholm
Norwegian Embassy in Helsinki
Norwegian Embassy in Moscow
Norwegian Consulate General in Murmansk
Swedish Embassy in Oslo
Swedish Embassy in Helsinki
Swedish Embassy in Moscow
Finnish Embassy in Oslo
Finnish Embassy in Stockholm
Finnish Embassy in Moscow
Russian Embassy in Oslo                                                                                                 Swedish Embassy in Helsinki
Russian Embassy in Stockholm
Russian Embassy in Helsinki
Russian Consulate General in Kirkenes

Regional government (links lead to English pages where available):

Nordland county, Norway
Troms county, Norway
Finnmark county, Norway
Norbotten county, Sweden
Västerbotten county, Sweden
Regional Council of Lapland, Finland
Council of Oulu Region, Finland
Murmansk county, Russia (only Russian)
Arkhangelsk county, Russia (only Russian)

Other Public Institutions and Organisations in the region:

The Barents Euro-Arctic Council
The Norwegian Barents Secretariat