Armed thugs had attacked protesters at the headquarters of the central bank in the capital, Abuja, it said.
For their part, police accused people “posing” as protesters of looting weapons, and torching police buildings in southern Edo state.
Prisoners were also reported to have escaped from a jail in the state.
The protests started about two weeks ago to demand an end to police brutality, with mainly young people using the social media hashtag #EndSars to rally people to demand the closure of the notorious Special Anti-Robbery Squad (Sars).
The unit, formed during military rule in 1984, was accused of extortion, torture and murder.
The protests have also been backed by global celebrities such as Twitter founder Jack Dorsey, US rapper Kanye West, footballers Mesut Ozil and Marcus Rashford as well as Nigerian superstars Davido and Wizkid.
The government agreed to disband the unit, but protests have since snowballed into calls for wider reforms with protesters now also using the hashtags #EndBadGovernance, #BetterNigeria and #FixNigeriaNow to build support on social media.
There was an increased military presence in Abuja on Monday – a day after Defence Minister Bashir Magashi warned protesters against “breaching national security”, the BBC’s Nduka Orjinmo reports from the city.
In Lagos, the country’s commercial hub, protesters blocked access to the main international airport by barricading the road.
Amnesty said police brutality has continued, with a 17-year-old teenager, named only as Saifullah, dying in police custody in northern Kano state on Monday.
She had allegedly been tortured to death, prompting protests in Kano’s Kofar Mata area, Amnesty added.
Police have not yet commented on the allegation.
In other developments, Amnesty said police fired tear gas at peaceful protesters in Abuja, while armed thugs had early on Monday morning attacked activists who had been occupying the central bank’s headquarters in the city.
Dozens of protesters were severely injured, the rights group said.
It was the latest in a series of “escalating violence and coordinated attacks” against #EndSars protesters, Amnesty added.
The attacks have also led to casualties in recent days in the cities of Benin in the south, and Oshogbo in the middle-belt, it said.
“We again call on law enforcement agencies to investigate these incidents and protect the protesters from further attack by hoodlums,” Amnesty added.
Meanwhile, the Edo state government has declared an indefinite curfew because of “incidents of vandalism and attacks carried out by hoodlums in the guise of #EndSars protesters”.
Earlier on Monday, prisoners broke out of a jail in the state.
Local media reported that more than 200 prisoners had escaped, but there has been no independent confirmation of this.
In videos posted on social media, some men could be seen scaling a high barbed-wire fence said to be the prison walls.
Police said two police stations and a police post were attacked in the state by “persons posing as #EndSars protesters”.
“The extent of damage cannot be ascertained at the moment but report indicates that the protesters carted away arms and ammunition from the armoury and freed the suspects in custody before setting some of the facilities ablaze,” it said.
Some Twitter users responded to the claim with scepticism.
“Nigerian police that are happy to shoot at peaceful protesters suddenly froze when prisoners are escaping??” one tweeter commented.