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If possible, you should request that your neighbor trim the branches. If you trim them and somehow the tree coincidentally dies, there would be further problems.
Trim the magnolia during the dormant season, which is winter. If you feel you must trim it, the branch must be removed at the trunk. Cutting pieces of branches can cause undue stress on the plant, causing illness or death.
You cant and you don't need to!
It will be cut off
cut the tree off at the ground and cover with concentrated glyphosphate
The best time to trim magnolias is in the summer and then agiain tin the winter. When trimming shape the plant in an overall cone shape, careful to cut the branches at a point where the next leaf is pointing away from the trunk and middle of the plant. This will ensure that the new "flush" of growth fills out the best. Blooms will occur in spring and will be maximized by another proper winter trim.
They should be trimmed from within
The neighbor. He is responsible to maintain the tree, but you are usually allowed by law to trim any portion of the branch that encroaches onto your property if you wish.
A pole tree trimmer is just what you need, you can find it at almost any hardware store.
After it has bloomed in the spring or while the weather is still cold in late winter - before the sap begins to travel to the branches. Make clean cuts, but, be aware that when you trim a large branch it may put out new sprouts that grow straight up. This can be used to your advantage if you want to create an interesting growth. Also know that the "bush" you refer to is probably a tree. If it is a Southern Magnolia it can become huge. If it is a tulip magnolia you are still looking at the possibility of a 20' tall and wide tree. Either way they are beautiful and smell great.
It is important to prune and trim trees of varieties. The main reason to trim a tree is the remove the dead and dying branches and lets the trees resources be sent to the living part of the tree and it helps promote new growth. Winter is the best time to prune because with the removal of the dead branches that tree is ready for spring.
Just remove the lower branches that are in your way. No other pruning is necessary.