The deputy has been placed on administrative leave while the incident is being investigated, but Virginia State Police told CNN Brown was unarmed. He has serious but not life-threatening injuries.

Brown's death has sparked both protests and widespread calls for the officers' body camera footage to be released publicly, including from North Carolina Gov. Roy Cooper, a Democrat. The district attorney for the region and the Pasquotank County attorney said in a joint statement Thursday that the body camera footage cannot be released without a court order.

In part because of that political polarization, there is still broad disagreement between the two parties about how far Congress should go to punish police misconduct.

The most recent shootings illustrate how the George Floyd Justice in Policing Act will only address some aspects of the problem, and would not necessarily have prevented any of the violence that unfolded last week. The bill has already passed the House, but has faced a more difficult path in the evenly divided Senate, where Democrats lack the votes to overcome a Republican filibuster.

Still, there are new signs of optimism that Republican and Democratic lawmakers are serious about trying to make a deal. Bass says she hopes the two sides can put together a framework by late May, which would be the one-year anniversary of Floyd's murder. Scott floated a potential compromise last week on reforming qualified immunity, arguing that police departments could be held accountable even if individual officers are still shielded. The South Carolina Republican has said some Democrats he has spoken with are open to his compromise and he doesn't believe Republicans are far apart on the issues.

GOP Sen. Lindsey Graham of South Carolina said Sunday that he believes there is a way to find compromise on qualified immunity.

"We can solve the issues if there's will to get there, and I think there's will to get there on the part of both parties now," Graham told Fox News' Chris Wallace on "Fox News Sunday."

But a number of progressive Democrats are voicing their concerns that Scott's compromise on qualified immunity would not go far enough in holding police officers accountable.

On Sunday, progressive freshman Rep. Cori Bush of Missouri questioned why police officers are entitled to a "safety net" -- in the form of qualified immunity -- when other professionals who deal with life-or-death situations do not have the same protection.

"The safety net shouldn't be there," Bush told CNN's Abby Phillip on "Inside Politics." "Where are all of the special protections for nursing and for other people in other positions that do very dangerous work?"

"We compromise on so much. You know, we compromise, we die. We compromise, we die," Bush added when asked about the compromise on qualified immunity floated by Scott, which she doesn't support. "I didn't come to Congress to compromise on what could keep us alive. ... If you don't hurt people, if you don't kill people, if you are just and fair in your work, then do you need the qualified immunity anyway?"

Rep. Stacey Plaskett, a Democratic congressional delegate for the US Virgin Islands, explained the passion behind the progressive push to change qualified immunity during an interview with CNN's Pamela Brown on Saturday night.

"Qualified immunity has in many instances become the hood for bad police officers to, in fact, act as modern-day Ku Klux Klan members against Black and Brown people in this country. And it has got to stop," Plaskett said. "The most conservative members of the Supreme Court say that Congress needs to do something about qualified immunity. And we cannot shirk our responsibility to victims and Americans at large because we are afraid of the unions, or talking points, or those on the right who have used the blue wall as a shield against American justice."

Biden administration steps up its visibility on the issue

Biden plans to make a push for the George Floyd Justice in Policing Act in his first address to a joint session of Congress on Wednesday, marking his first 100 days in office and laying out his priorities going forward. Newly confirmed members of his Justice Department are also taking a more active role on the issue.

Attorney General Merrick Garland, Deputy Attorney General Lisa Monaco and Associate Attorney General Vanita Gupta met in person at the Justice Department, and virtually, with police chiefs from major cities and influential police leaders from around the country Friday to discuss ideas for police reform, according to a spokesman for the attorney general. Garland also announced last week that he was opening a federal civil investigation into policing practices in Minneapolis.

When asked by CNN's Dana Bash in an interview whether she planned to get more personally involved in brokering a compromise, Vice President Kamala Harris implied that the onus is still on Congress.

"We've made our position clear, each of us," Harris said, referring to Biden and herself, "and as an administration we've made our position clear. But it is for the folks in the Senate to work together, to resolve whatever may be differences of opinion about the details of the legislation."

"I think there's no question that the American people, in a bipartisan way, realize and want that there will be some reform of the system," Harris added.

Her comments once again underscored that it is not yet clear how much political capital the White House is willing to spend to help draw Senate Republicans on board to reach a compromise -- and create a real chance for substantive change.

This story has been updated with additional details Sunday.