In his official report of the inquiry into unethical behaviour by a number of British newspapers, Leveson ruled that an independent watchdog should oversee UK press conduct and should be backed up by government legislation.
Following publication of the report on Thursday (November 29), inquiry witnesses Gerry McCann and Christopher Jeffries launched an online petition - run by the campaign group Hacked Off - which pushes for implementation of such laws, reports BBC News.
McCann - who faced "unbelievably damaging" newspaper reports suggesting that he and his wife killed their daughter Madeleine after she went missing in Portugal in 2007 - said that he did not think the report had gone far enough.
"Clearly the public want it, there's been a judicial review and I think the recommendations should be implemented," he said. "There's no good reason why they shouldn't be.
"The press has been given enough chances, and in my opinion Lord [Justice] Leveson has given them another chance to put a structure in place which they are happy with."
Jeffries - who was wrongly arrested for the murder of his tenant Joanna Yeates in 2010 - added: "Certainly I think [Cameron's decision to not adopt recommendations] has been influenced by the pressure he has received from newspaper proprietors and editors and by some MPs in his own party."
Labour leader Ed Miliband also made it clear that his party supports the campaign to introduce new press laws and described how the victims of press intrusion felt "utterly betrayed" by the prime minister's opposition.
"I am going to stand up for people... who have been appallingly treated by sections of the press and who put their faith in David Cameron, put their faith in the Leveson Inquiry, and who are frankly, I think, astonished by what the prime minister has done," Miliband said.
Parliamentary talks about the propositions began on Thursday night and are set to resume next week. The Department for Culture, Media and Sport is due to draw up a draft bill implementing the Leveson recommendations.
The prime minister is said to believe that the process will highlight how difficult it is to try to legislate such a complex and controversial area, while the Liberal Democrats - who have pledged to ensure the legislation is drawn up "in good faith" - are siding with Labour.
Chief political correspondent for the BBC News Channel Norman Smith has said that Labour sources fear the Conservative party will produce draft legislation deliberately written in a way that discredits the proposals - "like something the Stasi [East German secret police] had written".
However, Culture Secretary Maria Miller told the broadcaster that "the gauntlet has been thrown down" to newspapers who have first been asked to outline how they would approach self-regulation.
"[It] would actually give the opportunity in the future to bring into question the ability of Parliament to stay out of the issue of free press and difficult for Parliament to not have a statutory framework on which they could hang further bits of legislation," she explained.
Miller added that it is now up to the press to act "swiftly" and put in place a self-regulatory body that "adheres to the Leveson principles".