After pasting an object I make sure only one is selected, but both the original and the copy move together. I can enter edit mode and move one but they remain joined and ctrl+p to separate does not work. I'm not sure what has changed but I used to be able to copy an object and paste it as an entirely separate object.
When you duplicate objects be sure you are in object mode. If you are in edit mode they may conjoin and be made a single object. I have trouble with this too.