The 54-year-old former Manu Samoa and Auckland rugby representative and Ponsonby club member suffered a heart attack while travelling to a radio interview on the main island, Upolu.
It is understood members of the public came to his aid and called emergency services, but he did not make it to hospital.
Fatialofa regularly travelled to Samoa, where he had recently started Fats Tours,a private tour guide company.
He was also the owner of the Auckland-based Fats Enterprises, a successful piano-moving business that has been around for more than 20 years.
Fatialofa's family in Auckland were making arrangements to bring his body back for his funeral services.
A family spokesman said they hoped to bring him home by the weekend. A funeral is likely late next week.
Brother-in-law Poutoa Papali'i said Fatialofa's wife, Anne, and their eight children - Jeremiah, Vasa, Italia, Courtney, Manu, Shelby, Giovanni and Robert - were struggling with the news.
"We're being there for Anne and the kids, who really miss their father, and they're struggling. We're just here as an extended family to be here to support."
Mr Papali'i said that by yesterday afternoon, the family had received dozens of phone calls and messages of support.
They acknowledged the impact Fatialofa had on the community - not only through rugby and sports, but also as a Pacific Island and Samoan community leader - and were therefore planning to give him a fitting send-off.
"I had to remind my sister that this is going to be bigger than just the family because of what he's done for Samoa and rugby. You can't disassociate Fats from what Manu Samoa rugby is now."
One of Fatialofa's daughters, Courtney, said friends had contacted their mother early in the morning with the sad news. "It was sudden. He wasn't sick or anything."
She said their mother would fly to Samoa to bring their father home.
The family were struggling to find a way to tell Fatialofa's elderly mother about the death, as she is unwell.
From the Ponsonby Rugby Club, Fatialofa played 71 games over eight years from 1984 in the front row for Auckland.
Auckland Rugby Union board member and former teammate Gary Whetton said: "He was the ultimate team man - on the field he was huge and off the field such a gentleman ... a brilliant man."
Fatialofa played for Manu Samoa until 1996 - the same year he was made a Member of the New Zealand Order of Merit for services to rugby.
Tributes from around the world and rugby circles poured in. There is talk of a memorial service in Samoa.
Fatialofa captained Manu Samoa in their first Rugby World Cup appearance in 1991 and held three chiefly titles.
Tributes to Fats
All Black great Bryan Williams: "It's not just about Samoan rugby or Ponsonby or Auckland rugby. He transcended all boundaries. He was universally popular throughout the rugby world."
Former All Black Michael Jones: "He really was part and parcel of our community - especially in sports. I think Peter Fats embodied everything that was really special and unique about Samoan rugby.
He was a key figure who really brought Samoan rugby to the world and they became the darlings of rugby. He was a big man, with a big heart and a big spirit."
Former Manu Samoa player and All Black Frank Bunce: "He's so humble. His personality saw him being able to talk to politicians, the rich and famous and even to a man walking down the street with nothing. Fats was that character. He never treated anyone differently."
Former All Black Andy Haden: "I thought he was desperately unlucky [not to become an All Black] and he got very close. He was the loyalest of team men. He had a great affinity for Ponsonby rugby and then became a great member of the Auckland side of the '80s.
"Then when the opportunity came up in the World Cup and Manu Samoa came along, they couldn't have found a better captain and talisman."
NZ Samoan consul-general Fa'aolotoi Reupena Pogi: "He's very popular in Samoa and everyone knows him. You always saw him with a big smile and a bottle of Vailima [local beer] in his hand. I'm going to miss him and his jokes."
