Version: 2.81 build Oct 23 (Linux 64)
When a Curve Object Modifier uses the "Apply On Spline" setting, it seems to affect the curve data directly. This means that if a curve data block is used by multiple objects, a modifier on one of those objects will affect all the others.
To reproduce, open and inspect the attached file. Notice that the objects on the left both have deformation modifiers, while their clones on the right do not. However, the curve object still appears to be modified because it uses the same data as the modified curve object. This is not how modifiers are supposed to behave.
The expected behavior is demonstrated by the two mesh objects. Like the curve objects, they also share the same data, but the effect of the modifier is limited to the owner of the modifier.