The Independent Office for Police Conduct (IOPC) announced on Sunday it had opened an investigation into the incident in Summers Street.
Wiltshire Police officers were at the scene shortly after 2am on Sunday when police responded to reports of two men arguing in the street, the watchdog said.
On Monday, the watchdog revealed further details from its investigation, saying a firearms officer was first to attend the scene in an unmarked police vehicle, closely followed by two other officers in a marked police car.
The 57-year-old man was then shot in the chest by the firearms officer. He was declared dead in an ambulance at 2.56 am.
Initial accounts from the three officers and other relevant police personnel have been obtained, the IOPC said, and are being reviewed along with body cam footage.
It said it had verified that one shot was fired by the officer from a Block handgun.
A non-police issue firearm was recovered from the scene and will undergo forensic examination.
“We have had initial contact with the man’s family to explain our role and I send them my condolences for their loss, and my thoughts are with everyone affected," IOPC regional director Derrick Campbell said.
"We are at an early stage of our investigation and will be reviewing body-worn video footage from the officers involved as part of our enquiries.
"It is right where lethal force has been used that we carry out an independent investigation to carefully examine the circumstances and police actions.”
No arrests have been made.
Wiltshire Police confirmed in a statement on Sunday that a referral had been made to the IOPC concerning the death of a member of the public.
The force's statement added: "The IOPC are independently investigating this incident, so it would be inappropriate for us to comment any further on it.
"We would, however, like to reassure the public there is no risk to the wider community and that there is likely to be an increased police presence in the area for a considerable time."
Assistant Chief Constable Deb Smith said Wiltshire Police are continuing to assist the IOPC's investigation.
She added: "We would like to thank the residents of Summers Street and the wider community for their patience as inquiries continue in the area.
"We appreciate this will likely have caused concern and disruption for many people.
"However, this work is crucial as the Independent Office for Police Conduct (IOPC) continue to undertake its inquiries to establish the full facts of what happened this morning."
The chairman of the Wiltshire Police Federation called for a prompt investigation by the watchdog.
Mark Andrews said: "Officers who volunteer and are authorised to carry firearms on behalf of society, and who bravely do so to protect the public, are highly trained and are some of the best colleagues we have.
"Our thoughts are with all involved in this incident in Swindon and we are offering our colleagues our full support at this time as they go through the post-incident process.
"We call upon the Independent Office for Police Conduct to bring their investigation to a prompt conclusion for the benefit of all concerned.
"It would be inappropriate to comment further at this time due to the ongoing investigation."