Both bombers struck the communications department in the provincial capital, Sharana, Khaama Press reported. One attacker set off an explosive-packed vest inside and the other attacker detonated his explosives at the gate.
A security official said seven of the dead were insurgents and two were police officers.
Two bombs went off in Jalalabad, the capital of Nangarhar province, near the department of public works. The first caused no casualties, while the second wounded three people, officials said.
No one took responsibility for the bombings, although local investigators blamed them on the Taliban.