I would think it would depend on the situation....even though there's certainly a law in your area.
We have neighbors' trees fall onto our property every once in a while....but we're talking close to 200 acres....what's one more tree? We cut it up to the property line/fence, we don't go over on the other side to cut up theirs, nor do we throw any of the debris or punk over. We do step on the other side to fix the fence, but that's all.
I suppose if we thought about it, fixing the fence that it took down would be their responsibility. But, by the time we got ahold of them and nitpicked a few dollars out of them for the fencing materials/time....most of the herd would be out the hole grazing their land.
If it were up by the house and was our one neighbor's tree....AND....I KNEW they would/could use the wood.....I'd probably ask first....but would still clean up what's on our property, if they didn't want the wood.
I'm a bit confused to the postings suggesting that cleaning up a tree that's on your land is like keeping someone else's possesion.