My neighbors have very messy weeping willows along our shared property line. Who should be picking up all the limbs that fall every time the wind blows?

The person who owns the yard where they fall.

Seriously, can you imagine trying to identify every leaf that falls in your yard, tracking down the tree owner, and trying to make them pick them up?

I have not a single tree in my yard, yet, I'm the only yard on the block that has a fenced yard. Guess who ends up with ALL the leaves to remove every year.

But, I also know that some of those trees, block the afternoon sun, and keep my home much cooler than it would be without them.

It is the price you pay, to have your own place, if it bothers you that much, you can always move to an apartment. No yard, no raking, no neighbors with willow trees, and no privacy or place to call your own.

If the limbs of the trees are on your property you have the right to cut what is on your side. That may lessen the amount that you have to pick up.
I have no trees in my back yard but my neighbors on both sides sure do and we have a huge amount to pick up each fall in addition to them being old and losing branches. So I feel your pain, but I am a tree hugger and would never want them removed.

You can of course trim back any tree on your side of the property. But if the limbs fall on your side you pick them up

Unfortunately you have to pick up if dropped on your property, however instead of filling your bin with your neighbours garden waste, I'm sure it will give you great pleasure in returning all tree limbs and various bits of weeping willow to the neighbour in question.

This you have a perfect right to do as legally the bits of tree remain their property by law.

I'm sure after several returns, they will get the message and maybe have the tree trimmed to it's former glory.