Districts can raise tens or hundreds of millions of dollars via bonds (requires voter approval but usually passes). They can also pass parcel taxes, which aren't subject to Prop 13 although they can be regressive. Most of their money does still come from the state, but IMO that's not a bad thing, since keeping it more local raises issues with income inequality.
There's very few limits on how they spend their money. I think the only caveat is that districts get extra amounts of ADA money based on the attendance of certain categories of students (e.g. ones that are still learning English). They're supposed to spend that money on relevant programs, and the county has to certify that the spending on those programs lines up with the amount of extra money that is being received.
The local boards often do not have the power to:
(1) Raise revenue that would fund much independently of outside funding,
(2) Freely control how much of funding they do have is spent, because the outside funding sources or superior (e.g., state) law independent of funding sets narrow parameters for many of their decisions.