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One of the reasons I would personally never get in a transporter is that there is no evidence that the Real Me isn't just vaporized and a Copy Me created at the other location. The Copy Me might have all my memories and be indistinguishable from the original, but it isn't the Real Me.
I know that some early characters (especially in ST:Enterprise) had problems with transporters -- but these mostly seemed to concern safety. No one discussed the metaphysics of the technology.
Was there ever proof -- or even discussion -- of the question of whether the transporter really was transporting, or just creating a heck of a good copy?
Note 1: Yes, the transporter tech in ST sends the 'atoms' as well as the 'bits' to reconstruct the atoms, so it's not quite a 23rd century fax machine. But you're still being vaporized.....
Note 2: And yes, I do wake up each morning wondering if I am the same person who went to sleep the previous night, thank you very much. Why do you ask?