Yes, a custom piece can be reproduced. Should you ever have the misfortune of losing your custom piece, your jeweler should be able to reproduce it for you.
Can my custom piece be duplicated for others to wear? Most jewelers respect the fact that your custom piece was designed just for you, but whether or not your custom piece may actually be duplicated for other customers is solely a matter of business ethics. If you know and trust your jeweler, chances are he or she would never reproduce it other than with your permission or upon your request. However, you may want to have a conversation with your jeweler about this before ordering a custom piece, so your expectations are expressed and made known from the beginning.