In his first Downing Street press conference since returning from illness last Thursday, Mr Johnson appeared to signal a U-turn in government thinking on the issue when he described face coverings as “useful” because they give people the confidence to go back to work.
Government scientists are understood to think the evidence supporting face masks by the public is marginal.
Sir David told The Telegraph: “We want to help The Government, but there has to be an improvement in the public discussion.
“We need to show that there is a different way of doing things, that transparency is the way to win public trust, not having ministers endlessly saying that at no stage have they made any mistakes.”
The transparency of Sage’s decision-making process has attracted mounting disquiet in recent weeks, including the revelation that 16 of the 23 known members are employed by the government.
The current set-up allows ministers to justify their decisions based on “the science”, as advised by Sage, without explaining fully what the evidence says, Sir David said.
“”I am not at all critical of the scientists who are putting advice before the Government … but because there is no transparency the Government can say they are following scientific advice but we don’t know that they are.”
He added: “Science is fundamentally a system based on peer review. When it comes to scientific advice of any kind, transparency is essential.”
The rival committee will submit its findings to Downing Street as well as the Health and Social Care Select Committee.
Sir David has previously said the Government was too slow to impose the lockdown.
He has also criticised the attendance of Dominic Cummings, Mr Johnson’s chief political advisor, at Sage meetings.