Although the issue is devolved, until now there has a common system on both sides of the Border.
Writing in the Sunday Herald today, the First Minister says it is "unarguable" that there should be separate Scottish reforms, and suggests "light touch legislation" will be required.
But the three Unionist parties vehemently attacked the idea, with Labour even suggesting Salmond only wanted the means to "bully and intimidate the Scottish press" in the run-up to the independence referendum.
Salmond also signals that he will be at the heart of the process by confirming he will lead for the SNP Government in Tuesday's Holyrood debate on Leveson, despite opposition calls for him to step back because of his ties to Rupert Murdoch.
The First Minister's position means Holyrood, like Westminster, could struggle to reach a consensus on translating Lord Justice Leveson's report into action.
Salmond argues that Leveson has "thrown down the gauntlet to the Scottish Parliament" to devise its own system by acknowledging the different legal system in Scotland.
The First Minister also says he personally favours regulation underpinned by statute, in a variant of the system used in Ireland. He says features of the Irish model "crystallise the value of statutory underpinning", especially an independent ombudsman to handle complaints.
Salmond also questions Leveson's plan for the communications watchdog Ofcom to act as a "backstop" regulator for titles which refuse to join a new independent system of self-regulation, calling Ofcom "an extraneous complication".
Tuesday's debate will be the first formal opportunity for MSPs to air their views on the Leveson report since its publication last Thursday.
At Westminster, David Cameron immediately made it clear he opposed Leveson's plan for a new law underpinning press regulation, putting him at odds with his Liberal Democrat deputy, Nick Clegg.
The Prime Minister said legislation would amount to "crossing the Rubicon", warning that any law could be amended and developed by future governments to the point of censorship.
Calling on Cameron to prioritise the victims of press outrages, such as the parents of murdered schoolgirl Milly Dowler, and Gerry and Kate McCann, whose daughter Madeleine went missing in 2007, Labour leader Ed Miliband said he wanted Leveson's 1987-page report adopted "in its entirety". However, he has since signalled unease about plans to make journalists disclose information under data protection law.
North of the Border, Salmond is proposing a six-member independent implementation group, chaired by a judge with five non-politicians, to translate Leveson's broad ideas into practice. However, the way forward will only emerge after the cross-party talks, starting on Thursday.
In his article, the First Minister says the delay is "perhaps no bad thing given that, by moving too quickly, the UK parties have got into entrenched positions.
"While Westminster has split on the report recommendations, in Scotland we have the opportunity to reflect and find agreement."
Salmond stresses the difference – also underlined by Leveson – between statutory regulation and statutory underpinning of self-regulation. Under the former, politicians could influence newspaper content, while the latter means a new, independent system would be recognised in law to reassure the public it was effective, and would validate a code of ethics and handle complaints.
The Irish model consists of a 13-member Press Council with a majority of independent members, which appoints an ombudsman to handle complaints free of charge.
The ombudsman judges whether titles or individual journalists have breached a code of practice on accuracy, fairness and respect for privacy, and can refer serious cases to the Press Council. If complaints are upheld, titles must publish the Council's decisions with "similar prominence" to the offending story.
In return for signing up to the system, which is funded via a levy on its members, newspapers enjoy extra protection against libel actions. Until the regime began in 2008, Irish papers were loath to apologise in case a court took it as admission of liability in a libel case – they can now apologise without that inference.
Although the Press Council is recognised in law, it was not created by a law, keeping it independent of government.
Salmond said Scotland should not follow the Irish model exactly, but Holyrood should "look seriously at whether it can be adapted".
He added: "We should consider whether light touch legislation can provide maximum benefit and assurance of the protection of a free press in a sensitive regulatory environment.
"I believe it is clear that the case for a Scottish solution to these important and vital issues is unarguable."
However, any hope of agreement seemed remote yesterday, as opposition leaders said they were against the idea of a separate system. They also questioned Salmond's fitness to lead Thursday's talks, saying he was "tainted" by his eagerness to lobby for Murdoch's News Corp.
Leveson said Salmond had shown a "striking" willingness to lobby the UK Government on behalf of News Corp when it wanted to take over 100% of BSkyB. It said his rationale of saving Scottish jobs had been understandable but "irrelevant" to such a quasi-judicial decision.
Leveson concluded that Salmond "cannot be criticised" because he had never carried out his lobbying plans, but said if he had done so, it could have rendered a major government decision "unlawful".
Scottish Labour leader Johann Lamont said: "Alex Salmond has yet to make the case why for the first time Scotland needs a separate press regulator. I cannot see how it will benefit either the victims of press abuse, or the Scottish press. This looks like more grandstanding.
"Many will fear that he has manufactured this process so that he can bully and intimidate the Scottish press in the run-up to the referendum."
And Scottish Conservative leader Ruth Davidson accused Salmond of jumping the gun with his own plan before Thursday's talks.
"Newspaper publishers do not want and will not pay for two systems when one will do. The industry will sign up to a UK-wide system and if Mr Salmond truly believes in press freedom he cannot compel newspapers to join his system. Therefore, whatever he devises is likely to be redundant before it has begun."
Willie Rennie, leader of the Scottish Liberal Democrats, said the parties must unite around a single regulatory regime for the UK.
"We need cross-party consensus, but we also need cross-Border co-operation. With so many publications with Scottish and UK editions it makes sense that we agree on a press regulation regime that can be applied across the UK."
''Alex Salmond has yet to make the case why for the first time Scotland needs a separate press regulator"