Friends say Travaglini fought off the shark before it returned and dragged him underwater, the West Australian reports.
Opposition Leader Mike Nahan said the attacks emphasised the need to do more, including using SMART drumlines rather than the WA Labor government's subsidised deterrent devices.
It has been reported by local media that he was "conscious and breathing" during the rescue.
He has since entered the operating theatre for surgery on his wounds, and has been transferred to the State Trauma Unit.
St John Ambulance said the man in his 30s was being treated for leg injuries.
A member of the public reported an unknown species of shark off Cobblestones just moments after the attack, and the Surf Life Saving Westpac helicopter reported a four-metre shark just after 10am near the Lefthanders surf spot.
The Margaret River Pro global surfing competition, which is about 15km away from the attack, was suspended for about one hour, but then resumed at Main Break with Kiwi Hareb one of the two surfers who went back in the water first.
'Highly suspicious': Sydney bushfire remains out of control
The fire, which has already blackened 2,430 hectares, is now burning in a south-easterly direction towards Barden Ridge . Waterbombers were on the scene throughout the afternoon, however windy conditions are hampering firefighters's efforts.
IRS Awards Tax Filing Extension for Santa Barbara County
Still, he said, even if a person can not pay everything he or she owes, it's better to file it that day than not file it at all. So even if you owe more than you can pay right now, file your return and contact the IRS to set up a payment plan.
Boston Celtics vs. Milwaukee Bucks - 4/17/18 NBA Pick, Odds, and Prediction
The Greek Freak has scored at least 28 points in every game against Boston in 2017/18 to average 33.5 points on 53.9% shooting. But then Middleton responded after a timeout with an insane 35-footer off the inbound to force overtime.
"I$3 saw the guy who had been attacked get separated from the [surf] board and then start to paddle for an inside wave, which he managed to body surf all the way in", he told the ABC.
'They got him to shore and started working on him to stem the bleeding'.
"Sharks are still a threat to surfers and people in the ocean here", he told reporters.
A World Surf League spokeswoman later said the event was back up and running.
"The safety of our surfers & staff is a top priority".
A spokeswoman said the comeptition wanted to make sure that the surfers were protected.
Nick Edwards, 31, also died after a shark attack while surfing at nearby South Point in 2010. Organizers added more drones and jet skis to ensure the safety of competitors.