Coulson, 45, is accused of requesting and authorising payments to public officials in exchange for information, including contact details of the royal family.
He is charged alongside Clive Goodman, the former royal correspondent for the News of the World, the tabloid which was shut down by Rupert Murdoch in 2011 amid a wave of public revulsion over phone hacking.
Brooks, 44, a journalist and former editor who rose to become chief executive of Murdoch’s British newspaper unit News International, is also accused of making payments to a public official.
The two separate cases were adjourned until next month.
The police investigation into allegations of bribery at the News of the World and other newspapers is running alongside a probe into phone hacking.
Scandal rocked Murdoch empire
The hacking scandal rocked Murdoch’s media empire and embarrassed Cameron, who is friends with Brooks and also hired Coulson as his communications chief.
John Kay, the chief reporter for Murdoch’s The Sun tabloid, meanwhile pleaded not guilty on Friday to conspiracy to commit misconduct in public office.
Separately, a former police officer, Alan Tierney, admitted selling information to The Sun about the arrests of footballer John Terry’s mother and Rolling Stone musician Ronnie Wood.
The information concerned Sue Terry and Sue Poole, the mother and mother-in-law of the former England football captain, who had been arrested on suspicion of shoplifting, and were given police cautions.
Wood was arrested on suspicion of beating up his Russian lover Ekaterina Ivanova. He also accepted a caution.
Another public official, prison officer Richard Trunkfield, 31, also admitted in court on Friday to leaking information about a high-profile inmate to The Sun.
In two other separate cases, two officials admitted misconduct in public office. The court banned the publication of any further details about them.