This is working as intended. The same widget can not be in two places at once, additionally each widget can have different state. For example, imagine a case where you show friendly names in Green and Enemy names in Red. The same widget could not produce this effect, so instead you need separate widgets per player screen.
If you need to use the widget component in a split screen scenario, I’d recommend spawning multiple copies, and assign each one to the player it needs to belong to. You can do this by calling SetOwnerPlayer on the component, that will place it on that player’s screen.
If this is a case you run into a lot, I might recommend making a WidgetComponentSpawnerComponent or something like that, that automatically creates widget components for each local player, manages lifetime…etc.