One witness said: "Suicide car bombs targeted the presidential palace and the Ethiopian embassy.
"I saw smoke coming out of the presidential building. I also saw one dead soldier in front of the gate, but I could not get closer."
One police official said an employee of Ethiopia's embassy was badly wounded in the blasts.
There was no immediate claim of responsibility for the attacks in the two cities, which had been largely spared the fighting that has rocked southern and central Somalia in recent months.
A former British protectorate, Somaliland united with the Italian Somalia in 1960. It unilaterally broke away and announced independence 10 months after Mohamed Siad Barre, then president, was removed from power in 1991.
Muse Gelle, the governor of Bari in Puntland, said two blasts hit the offices of the Puntland Intelligence Service in Bosasso port on Wednesday.
"Two suicide bombers exploded cars in the Puntland Intelligence Service (PIS) compound," Gelle said.
Mohamoud Musa Hirsi Adde, the president of Puntland, said six members of PIS were killed in the explosions.
According to witnesses, the explosions took place at about 10.30am (07:30 GMT).
Puntland declared itself autonomous from the rest of Somalia in August 1998 under the leadership of Abdullahi Yusuf Ahmed, the current president of the Somali interim government.
The attacks took place as leaders of Somalia's UN-backed interim government met regional heads of state for talks in Nairobi.
The four-year-old administration is under pressure to end the chaos and share power with moderate opposition figures.