Laws vary from location to location. In the vast majority of cases, you have the right to remove the part of the tree that grows over into your yard as long as the pruning you do does not cause the tree to fail. However, there are exceptions to this.

Having a right to do something is different than having the responsibility to do it. I don't know for sure if you really meant responsibility, but, if you did, I don't know of any reason to believe that your neighbors would have the responsibility to remove a limb growing into your yard.

We split the cost with the neighbor to have their tree limbs that shaded half our back yard trimmed, took a year for their half to be paid to us but in the meantime we got some sun on that side of the yard and allowed new plantings to grow. Have a chat with your neighbor and see what you can work out.

You can only cut the branches that are over your property line, which means unto your property line. Ask for permission and see if they will allow you to have the limbs removed by a professional to ensure proper removal so as to not kill the tree.