Dividing fence matters are regulated by the Neighbourhood Disputes (Dividing Fences and Trees) Act 2011. Under the Act:
- Adjoining owners are liable for contribution to fencing work to create a sufficient dividing fence; and
- Fencing work includes replacement, removal, repair and maintenance of the whole or part of the dividing fence.
I can't see anything in the Act that makes the issue different for particular types of fences, such as asbestos.
The Act sets out what a sufficient dividing fence is. In most cases it is:
(a) Between 0.5m and 1.8m high (but it can be higher, really);
(b) Substantially made of a prescribed material, being: wood, chain wire, metal panels/rods, bricks, rendered cement, concrete blocks, hedge/vegetative barrier, or "other material of which a dividing fence is ordinarily constructed".
The thing with asbestos, at least in Queensland, is that it doesn't have to be removed unless it becomes damaged - it is considered not to be a risk to health while it is in a 'bonded' state. Removal of asbestos is regulated and must be done in the approved manner, and liability for the removal of asbestos rests with the owner of the property.
So, providing the fence is in good, undamaged condition, and given that asbestos bonded material were commonly used in construction at the time the fence was constructed, it is likely that the fence will be considered a 'sufficient dividing fence' in its current state.
If that is the case, then then neighbour has to bear the whole cost of removal and replacement because the fencing work isn't necessary.