If the trunk of a tree touches your and your neighbors' property, it is considered a shared (property-line) tree.
In general, you are responsible for any work done on your side, and your neighbors for any work on their side. However, since anything done on your side of the property line will likely impact the overall health of the tree, it's always best to clear your intentions with your neighbors before getting the work done. You certainly don't want to wind up with a nasty situation in which your neighbors hate you (or sue you!) because the tree service damaged or killed the tree. Give them a chance to provide some input!