Q: The trees on our lot have branches that overhang a neighbor's yard, and she is insisting that it's our responsibility to trim them. Is that true?
A: It's generally accepted that if a neighbor's tree overhangs your yard, you can whack it back to the property line. The matter of who pays for it isn't so clear-cut, and, in a hostile environment, could wind up before a judge. That's likely to be far more expensive than hiring someone to do the work.
If you value your trees, you might want to take care of things yourself or at least share the expense of hiring an arborist. A hatchet job by your neighbor could damage and weaken the tree. It could even kill it, and then, next time the wind blows strongly, it could fall into one of your yards or onto one of your houses. That would undoubtedly lead to a lot more unpleasantness and expense.
The moral of this story is that it's in both your interests to work this out amicably. Better to be safe than to be sorry.