I live in a semi-rural area, but not far from the brickwork chimneys, so wood burners are the least of it.
However, I do have 16 trees which need keeping under control, so I am grateful that I have a neighbour who will burn my cuttings. We have a low height restriction on chimneys, so even though he is upwind of us, we never get any problem.
Given that my council will only take the larger pieces in general waste, so they end up in landfill.
The smaller pieces go into an aerobic digester for electricity.
If your neighbour's smoke is a regular nuisance, it is possible that it is not correctly installed, so it would be worth asking your council to check.