In fact, if you do not believe in transcendental souls of some kind, there should be no fear of this type of scenario. You anyway go to sleep each night and awaken after a discontinuity. Why would this be significantly different? Now, killing yourself after having your mind copied to the machine doesn't make any sense.
More interestingly though, if we had the necessary technology for this type of transfer, we should also be able to do mind/machine interfaces, such that either copy of you could directly experience what the other does, meaning that this could be experienced differently from a notion of either a single entity OR 2 separate entities. Of course, it would follow that in fact multiple people could share experiences in similar ways, with unclear effects on the very notion of identity.
If I could see, feel, hear, even think the same things as you, would the concept of me and you as separate identities actually mean anything to us anymore? I suspect that we can't really imagine what society and humanity would actually look like if/when such technology were widely available.