Justine Damond, a qualified vet and yoga instructor, was remembered as an "angel on earth" and "a guiding light" by loved ones.
The 40-year-old, also known as Justine Ruszczyk, was shot dead in Minneapolis about 11:30pm local time on Saturday. The officers involved did not have their body cameras turned on.
"We are trying to come to terms with this tragedy and to understand why this has happened," her family said in a statement.
Matt Omo, from Manly, was a friend and colleague of Ms Damond, and had known her for about 10 years.
"I only hope this evolves into something that can make a positive impact for the world," he said.
"How someone teaching meditation and spreading love can be shot dead by police while in her pyjamas is beyond comprehension."
Ms Damond went to Manly High School and graduated from the University of Sydney.
She was to marry Minneapolis local Don Damond next month.
Julia Reed, who has known Ms Damond for 32 years, said her friend had brought intelligence and joy to life.
"She was treasured and loved, and we will really miss her," she said.
Alison Monaghan, who trained Ms Damond in alternative therapies, described her as "the most beautiful person" who was "all about giving to other people".
She said Ms Damond moved to the United States to "follow her heart" and she had a "new life".
Ms Damond's stepson Zach Damond, 22, said she called police after hearing a noise in the alleyway near their house in the suburb of Fulton.
'May your beautiful light shine through the darkness'
Many friends took to social media to express their grief, with one describing Ms Damond as "a guiding light, a dear friend and an incredible spirit".
"Thank you my love. Now a force even more powerful than before, may your beautiful light shine through the darkness," they wrote in a post.
"May your love radiate and heal all who now mourn your absence."
In a Facebook tribute, one woman said she spent many hours speaking with Ms Damond "talking about the world she was so determined to change".
"Justine was one of the most peaceful, calm, helpful people I have ever met," they wrote.
"So it's hard for me to wrap my head around why this would happen to her."
The Minneapolis Police Department said the incident would be investigated.