No it shouldn’t, though there is currently an engine bug with instanced inline-new objects where they can end up referencing the global version of an object. It appears to happen most often with grip scripts if you add one to a base class and then create a BP off of that base class.
There are a few reports in so I hope it gets fixed soon. How are you spawning them? I can dynamically spawn scripts at will and they behave independently as well as ones just placed around.