It is very simple. The tree-limbs overhanging the common property are the responsibility of the OC as they exist in your air-space (irrespective of whether the trees are growing on a neighbour’s grounds).
Therefore, if you wish to remove those limbs in your airspace, you may do so at your expense and offer to return the limbs to the neighbour as these are the property of your neighbour. You may not throw the limbs over the fence.
Ensure that there is no preservation order on the trees – either because of significant value or simple council tree preservation rules – or you could be fined if you lop branches. If your actions kill the tree you will be liable. I would recommend reading the Trees (Disputes Between Neighbours) Act 2006 (https://www.legislation.nsw.gov.au/sessionalview/sessional/act/2006-126.pdf)