The convention was responding to a stage one report produced by the Scottish Parliament's local government committee on the Local Government in Scotland Bill.
COSLA president Pat Watters (Lab) said the report reflected concerns raised by COSLA and the Society of Local Authority Chief Executives & Senior Managers on the draft bill.
'In particular, it supports our call for best value to be rolled out across the public sector - same duty, same guidance and, perhaps most importantly of all, same accountability arrangements,' he said.
The local government committee heard evidence from the Welsh Local Government Association, the Improvement & Development Agency and members of the public on its renewing local democracy inquiry.
Committee convenor Trish Godman MSP (Lab) said: 'Encouraging local people to take ownership of their areas is an important step towards renewing local democracy and we want to hear from people about how they are currently working towards this aim.'