Councils face a difficult task in raising standards in schools over which they have little direct control, Changing Partners says. And they must meet their targets while carrying out their other duties, from housing to lifelong learning - less headline-grabbing, but all laid down by law.
They are also the focus of an increasingly noisy debate which questions the whole role of local government in education - and whether it should have one at all.
The spotlight is particularly bright this month as the Audit Commission and the Office for Standards in Education embark on LEA inspections. The commission is also preparing a major report, due in the summer, which will pull apart LEA finance, examining how much cash authorities keep centrally and scrutinising their effectiveness.
But the speed of the Government's education programme - including the School Standards and Framework Bill already before Parliament - has prompted the commission to release a discussion paper to try to steer the debate.
Changing Partners says failure to meet the challenge of raising standards will call into question the very existence of local authorities. But fundamental questions need to be answered by government, LEAs and schools if they are to succeed:
* what is the authority's role?
* what makes an effective authority?
* how do we measure their performance?
* how can we ensure inspection helps rather than hinders LEAs?
* what powers is the Government going to give LEAs to help them carry out their role?
The report warns that setting targets for measuring an LEA's performance is harder than it looks - not least because schools have so much autonomy. There is also the danger of "plan proliferation" - authorities are already involved in preparing plans in a dozen areas, from early years to class sizes.
Greg Wilkinson, the commission's director of local government studies and the report's principal author, said: "We are trying to raise LEAs' sights and broaden their horizons.
"But we are also encouraging them and the Government to think about how difficult the task is, and how there is an awful lot LEAs will have to do if they are going to make a decent fist of it."
Changing Partners suggests the role of LEAs is a mixture of providing a vision, stimulating improvement, ensuring equity and acting as an arbiter when parts of the service come into conflict.
And it suggests a model of an effective authority based on partnership, with a clear strategy that gives it a strong focus, a coherent way of operating and a culture that brings the best out of staff.
The report warns some will call the model an "unattainable ideal for a democratically-elected public body" and admits not all LEAs will be up to the task.
But Mr Wilkinson said: "This is not a platonic ideal that's been dreamt up in some academy. This is something that has been informed by experience of authorities in the real world."