He said that plans to allow voluntary sector representatives to sit on local authority scrutiny committees should be reciprocated by "some degree of scrutiny of the voluntary sector" by councils.
Beecham, chair of the Local Government Association, made his comments at the fifth Compact annual meeting last week.
After the meeting, he told Third Sector: "A charity like the WRVS could talk to the council about its business plan and explain its policies in the same way that the Learning and Skills Council and utility companies would. This could even apply to councils for voluntary services."
Beecham, the former leader of Newcastle City Council, also said that there was a case for extending the scope of the Compact to include police and fire authorities and further education colleges, as well as local councils.
The Compact meeting was held in the wake of new research revealing that 89 per cent of the country is covered by a local compact or is developing one. There are compacts in more than 200 local authority areas, and 89 have been agreed in the past year.
Sir Michael Bichard, chairman of the Compact Working Group, said: "It's time to move the debate on from coverage to quantity. Organisations beyond local voluntary sector infrastructure groups need to understand how important the Compact is and how it affects them."
NCVO chief executive Stuart Etherington highlighted issues that had been raised with the Compact Advocacy Service.
He said the "old chestnut" of disproportionate government auditing and monitoring procedures for grants remained a problem. In one case, pay slips containing confidential information had been demanded by the statutory funder.
Last-minute notice of funding cuts and uncertainty about funding were also issues, such as in the Children's Fund affair, which Etherington said had left a "legacy of distrust".
David Tyler, chief executive of Community Matters, said there was a case for reviewing long-standing local compacts in the light of recent developments such as the Community Groups Compact Code, which was launched in July last year.
The action plan agreed at the meeting included a target to extend the Compact to all remaining local authority areas by April next year, and to increase the number of public sector bodies in local compacts. A revised funding code will also be published in the next 12 months.