Mr Salmond said he believed there had been "significant progress" on the issue in the last year but called for the creation of an independent recognition panel, as suggested in the Leveson Report, and an arbitration service.
Former solicitor-general for Scotland Lord McCluskey chaired a Scottish Government-appointed group to review the Leveson Report last year.
It concluded that the Scottish press should be subject to mandatory regulation underpinned by law but the Scottish Government rejected the proposal, instead supporting voluntary press regulation through a royal charter on press conduct and securing cross-party support for it.
The charter was passed through Westminster but the industry set up its own regulator, the Independent Press Standards Organisation (Ipso), as it said the charter may allow outside control of the press.
Mr Salmond was speaking at the first Scottish Newspaper Society conference in Glasgow.
He said: "My view is that Ipso will go actually a long way to meeting Lord Justice Leveson's proposals for independent self regulation, but it is important to set out the position of the Scottish Parliament.
"We're working, as a government, within a mandate that was set by a motion that was passed unanimously in the Scottish Parliament 12 months ago. That mandate required the Scottish Government to work with the UK Government to agree the text of the royal charter that implemented the Leveson recommendations in a way which was workable in Scotland.
"A key part of the recommendations was to have an independent recognition panel to assess whether the charter's requirements had been met.
"The Scottish Newspaper Society is considering how a new regulator can be scrutinised or questioned by a committee at the Scottish Parliament, I welcome the intentions behind that move.
"But the current proposals are different from agreeing to a recognition panel, and in my view are not as satisfactory. In fact, it could be argued that a recognition panel might be a better way of safeguarding press independence, because politicians by definition would not have a role in it.
"The royal charter also requires the independent press regulatory body should offer an arbitration service, and makes it clear that the Arbitration Scotland Act (2010) can underpin such a system.
"Under these proposals a recognised press regulator is required to offer an arbitration service, no one is required to use the service, arbitration by definition cannot be compulsory, but it would make available a cost-effective and relatively quick method of settlement.
"It would be especially valuable for people who feel they have been victims of defamation or breach of privacy and it would benefit newspapers by reducing legal costs and expenses."
Mr Salmond said he did not know any politician across the UK who was calling for state regulation of the press.
"There is a seeking of an underpinning of independent self regulation of the press," he said.
"The Scottish Government's basic view is clear; we strongly welcome that significant progress that has been made in the last year but we believe that the press should agree to the independent recognition panel and should provide arbitration services.
"These two remaining steps would be good for public confidence in the press, good for the press itself and therefore we hope that further progress can be made."
The First Minister also said there was no reason that sharing regulation across the UK could not continue after independence.
He added that the independence referendum is "the story of a lifetime" and that there is no regulatory or financial threat to the newspaper industry from a Yes vote.