As for why one beech is losing leaves and one isn't, it is usually all about energy production to get the tree through winter and into the coming year. If the canopy around a particular tree is a bit thicker or even if it is shaded by a feature of the terrain more than another, it will retain leaves longer in order to produce more food. Smaller immature beeches often don't drop their leaves until spring, even though the leaves are dead and brown. It is called marcesence. They hang on to leaves to produce the maximum amount of food until they can't produce anymore. The leaves then wither but remain until spring when new leaves emerge. I'd bet if you look closely, you will find that one tree is in a more favorable position for sunlight for one reason or another. That tree can afford to shed leaves in the fall, reducing wind load and jettisoning waste material while the other needs to produce food for longer.
As for why one beech is losing leaves and one isn't, it is usually all about energy production to get the tree through winter and into the coming year. If the canopy around a particular tree is a bit thicker or even if it is shaded by a feature of the terrain more than another, it will retain leaves longer in order to produce more food. Smaller immature beeches often don't drop their leaves until spring, even though the leaves are dead and brown. It is called marcesence. They hang on to leaves to produce the maximum amount of food until they can't produce anymore. The leaves then wither but remain until spring when new leaves emerge. I'd bet if you look closely, you will find that one tree is in a more favorable position for sunlight for one reason or another. That tree can afford to shed leaves in the fall, reducing wind load and jettisoning waste material while the other needs to produce food for longer.