Investigators returned Thursday to the downtown Montreal street where officers fatally wounded two men, while a renewed debate raged on how police shootings should be handled in the province of Quebec.

CTV's Montreal Bureau Chief Genevieve Beauchemin said the provincial police were not saying much Thursday about what they were doing back at the scene.

"For now they're very tight-lipped," Beauchemin told CTV's Canada AM on Thursday morning.

While the full circumstances of the shooting are not yet known, witnesses have said a group of Montreal officers were pursuing a knife-wielding homeless man when they opened fire on Tuesday morning.

When the shooting was over, the homeless man lay dead and an innocent bystander had been fatally wounded.

The homeless man has been identified as 40-year-old Mario Hamel.

The bystander was Patrick Limoges, a 36-year-old Montreal man who was on his way to work at a local hospital when he was shot. He was pronounced dead several hours after being hit by police fire.

The deaths of the two men prompted dozens of anti-police demonstrators to take to the streets Wednesday night, chanting slogans and carrying signs denouncing police violence.

Among the demonstrators' grievances was the fact that that the provincial police are handling the investigation into the Montreal shooting.

"There was another murder by police and the other murders that happened in the past were never punished -- I think that's a problem," demonstrator Jean-Luc Simard said Wednesday before the demonstration began.

"We know the same thing is going to happen again -- the Surete du Quebec (provincial police) will justify this assassination."

In Quebec, it is customary for an outside police force to handle an investigation into a police shooting.

But in the wake of Tuesday's double shooting, critics and protesters are saying that it is time for that tradition to change.

"For the first time I heard the public security minister yesterday kind of musing about changing the system even though this has been a point of contention for a number of years here in the province," Beauchemin said.

Official statistics from the Quebec government indicate that 72 people have been killed or seriously injured by police fire in the province over the past 12 years.

With files from The Canadian Press