In a statement the department said that it had taken the decision in light of the "exceptional circumstances" surrounding their 13-day ordeal in Iranian captivity.
But Lt Carman, the senior Royal Navy figure seized by the Iranians, told the BBC that he was "not interested in making money out of this".
"My main aim is to tell the story," he said.
"There's some people who might be making money, but that's an individual's decision, that's very private, but that's not something that myself or many of the others will do.
"I'd happily do it for free. But if there is money on offer then it would probably go to charity," Lt Carman added.
Royal Marine Captain Chris Air meanwhile echoed his fellow officer's comments by insisting he had no intention of selling his story to the highest bidder.
But defending the rights of the Royal Navy personnel to do so, he told ITV Granada News: "I'm not going to speak to the papers. I think some people are going to, and they are perfectly entitled to.
"I think it can be part of the process to get things off their mind. To be honest, it didn't seem that traumatic at the time to me and I don't think it's going to affect me in a terrible way."
Opposition parties have condemned the decision to allow the released sailors to sell their stories, with shadow defence secretary Liam Fox describing the move as "undignified".
"Many people who shared the anxiety of the hostages' abduction will feel that selling their stories is somewhat undignified and falls below the very high standards we have come to expect from our servicemen and women," he said.
Liberal Democrat defence spokesman Nick Harvey said that the MoD's decision would "certainly raise eyebrows".
"I sincerely hope this will not backfire into a loss of public sympathy for the service personnel," Mr Harvey added.
Explaining the government's decision to allow the group from HMS Cornwall to sell their stories following their release, the MoD said the decision had been taken as a result of "exceptional media interest".
"It is a fact that the media have been making direct contact with the families and offering them significant sums of money – this is not something that the navy and the MoD have any control over," the statement continued.
"Quite aside from the 'human interest story' surrounding these individuals, there are also sensitive strategic issues in play. It was clear that the stories they had to tell were likely to have emerged via family and friends regardless of any decision the navy took.
"It was therefore decided to grant permission to speak to the media to those personnel that sought it, in order to ensure that the navy and the MoD had sight of what they were going to say – as well as providing proper media support to the sailors and marines in the same way as would have been the case in more ordinary circumstances," the MoD concluded.
Reports suggest that the 15 may now be able to command up to six figure sums for details of their ordeal, with the Sunday Times and Sunday Telegraph claiming that Leading Seaman Faye Turney - the only woman among the group – has already sold her story to ITV1's Tonight with Trevor McDonald show and a newspaper.
The eight Royal Navy sailors and seven marines arrived back in the UK last Thursday after being released by Tehran, which had insisted that the group had been in Iranian waters when they were seized on March 23rd - a claim refuted by the British government.
In the diplomatic row which ensued, images of the imprisoned military personnel were shown on Iranian television and beamed across the world as officials in London and Tehran sought to find an end to the stand-off.
The plight of the sailors attracted further media interest on Friday, when one of the released service men told a press conference that he and his fellow captives had been blindfolded and stripped during their ordeal, during which time they "faced constant psychological pressure".
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