The police chief and the head of crime prevention for the southern Amhara district of Dejen were killed and a driver wounded in an attack on Monday, the local Dejen government said in a statement on Facebook.
On Tuesday, the police chief for the town of Shewa Robit was killed, the mayor's office said on Facebook, without giving further details.
The authorities said a curfew would be imposed for an indefinite period from 6:00pm Wednesday in Shewa Robit, which lies east of Dejen and around 200km northeast of the capital, Addis Ababa.
There has been no immediate claim of responsibility for the killings, which follow the assassination in April of a leading member of Prime Minister Abiy Ahmed's party.
Girma Yeshitila, the head of the Prosperity Party in Amhara, was among five people killed in what Ethiopian security forces said at the time was an attempt to overthrow the regional authorities.
Amhara nationalists have accused Abiy's government of seeking to dismantle the region's "special forces" in a bid to weaken the region.
The Amhara forces played a crucial role alongside the federal army during a brutal two-year war in the neighbouring Tigray region, which ended with a peace agreement in November.
Both Dejen and Shewa Robit lie close to the border with Oromia, the largest and most populous region in Ethiopia and where the Oromo Liberation Army (OLA) has been fighting a rebellion against the federal government.
Amhara has been the scene of unrest since a controversial move by the federal government earlier this year to dismantle regional forces and integrate them into the national army or regional police.
The people of Oromia and Amhara are also at odds with each other over regional border issues.