Channel 4 has called on Ofcom, the broadcasting regulator, to investigate its "cash-for-access" sting on two former foreign secretaries following criticism from MPs.
The Independent Press Standards Organisation, meanwhile, has received no complaints about The Daily Telegraph's coverage, either from the politicians involved or from any third parties.
The Parliamentary Commissioner for Standards cleared Sir Malcolm Rifkind and Jack Straw and said the "damage" done to the former MPs could have been avoided if Dispatches and The Daily Telegraph had "accurately reported" the exchanges they had filmed.
But the newspaper said it believed voters would find it "remarkable" that Parliament investigates the misconduct of its own members while Channel 4 issued a defiant statement defending its journalism and took the unprecedented step of asking Ofcom to look at the case.
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A Channel 4 spokesman said: "This programme raised important questions which concern voters about how senior politicians are able to use their public office for personal financial gain. This is a matter of public interest and was a legitimate journalistic investigation.
"We're confident in our journalism and have decided to take the unprecedented step of inviting our statutory regulator Ofcom to investigate it.
"Full transcripts of the interviews are in the public domain and we are in the process of making the programme available on All4, our online platform, to enable people to make up their own minds."
Downing Street said Prime Minister David Cameron welcomed the fact that Rifkind and his family can "now put this distressing episode behind them".
A No 10 spokesman said on Thursday: "The Prime Minister welcomes today's report by the Parliamentary Commissioner for Standards which has cleared Sir Malcolm Rifkind following allegations made earlier this year. The Commissioner makes clear that Sir Malcolm did not breach the rules on paid lobbying.
"After a long and distinguished career in the House of Commons, the Prime Minister welcomes the fact that Sir Malcolm, and his family, can now put this distressing episode behind them."
Undercover reporters claiming to represent a Hong Kong-based communications agency called PMR that was seeking to hire senior British politicians to join its advisory board secretly filmed the former MPs.
Rifkind was said to have claimed that he could arrange ''useful access'' to every British ambassador in the world because of his status, while Straw boasted of operating ''under the radar'' to use his influence to change European Union rules on behalf of a commodity firm which paid him £60,000 a year.
Commissioner Kathryn Hudson found "there was no breach of the rules on paid lobbying" or the rules of the House "other than in Straw's case – by a minor misuse of parliamentary resources".
She added: "If in their coverage of this story, the reporters for Dispatches and the Daily Telegraph had accurately reported what was said by the two members in their interviews, and measured their words against the rules of the House, it would have been possible to avoid the damage that has been done to the lives of two individuals and those around them, and to the reputation of the House."
In a report published by the Commons Standards Committee, MPs said Rifkind and Straw had been "scrupulous in observing the requirements relating to registration of interests".
The committee also raised concerns about media coverage of the allegations and claimed the former cabinet ministers had been "presumed guilty" before any investigation had taken place.
Rifkind, who stepped aside as chairman of the parliamentary Intelligence and Security Committee and quit as an MP at the election following the claims, said the months after the sting had been a "painful period" for him and his family.
He said: "Channel 4 Dispatches and the Daily Telegraph must recognise the judgment of the Standards Commissioner and the Standards Committee that they were responsible for 'distortion' and for misleading the public in making these allegations.
"It has been for me, for my family and for my former parliamentary staff a painful period which we can now put behind us," he added.
"My public life has continued over the last seven months with the support of colleagues. I am looking forward to the years ahead in very good spirits."
Straw said he was pleased with the outcome of the report and told how the allegations had "taken a huge toll" on him and his family.
He added: "At the time of this sting I said that I felt mortified that I had fallen into this trap but that I had not acted improperly in the meetings, nor more widely in respect of my Parliamentary duties and the rules of the House.
"I have been fully vindicated in this. The commissioner's report gives the full context of what happened, which was not available to the public at the time.
"It has been very sad that the final chapter of my long period in the Commons has been overshadowed in this way.
"The whole episode has taken a huge toll on my family, my friends, and on me, but the commissioner's conclusions and the committee's findings will now enable me to get on with my life."
Labour former home secretary Alan Johnson attacked the journalists involved in the sting.
He said: "Anyone who knows Jack Straw well recognises that he values integrity above all other virtues and that he has that quality in abundance.
"His distinguished parliamentary career did not deserve to end overshadowed through the antics of certain journalists for whom integrity meant little.
"This judgment lifts the shadow and ensures that the reputation of a brilliant politician remains intact."
A Channel 4 spokesman said: "Channel 4 Dispatches stands by its journalism; this was a fair and accurate account of what the two MPs said. This investigation was in the public interest and revealed matters which were of serious public concern."
A spokesman for the Telegraph said: "The Daily Telegraph conducted an investigation that was in the public interest and accurately revealed matters which were of concern to millions of voters.
"We raised a number of serious questions about the conduct of Members of Parliament. We suspect voters will find it remarkable that, despite the scandal of MPs' expenses, Parliament still sees fit for MPs to be both judge and jury on their own conduct."
Standards committee chairman Kevin Barron said: "What was said in the interviews should have been reported accurately and measured against the rules of the House. Jack Straw and Sir Malcolm Rifkind were presumed guilty before any authoritative investigation had taken place."
An Ofcom spokesman said: "We are aware of the Commission's report, and are reviewing the findings carefully. We will then assess whether any investigation is appropriate."
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