Pointers don't instantiate dynamic memory allocation. Pointers and allocations are entirely different things.
If you copy a pointer that points to dynamically allocated memory, you have two pointers pointing at the same allocated memory. Since you've copied it, it already points to the memory block. Specifically, if using the compiler-generated copy constructor, the new pointer will point to the exact same thing as the old pointer. You don't need to do anything with it if that's OK.
You do have the problem of when to free the memory. Freeing it twice will typically cause heap corruption, which is nasty. Not freeing it will cause a memory leak, which may be acceptable in some circumstances. Freeing it before the other pointer is through with it will also cause problems. For this reason, people who have multiple pointers to the same memory often go to the Boost project and use their shared_ptr template (which will be in the upcoming new Standard, and is present in most up-to-date systems).
If you want each pointer to point to separate chunks of memory, you have to set that up by writing your own copy constructor, and allocating a new chunk and copying necessary data in it. (You also need to write your own assignment operator, for the exact same reasons, and your own destructor so you can free the memory. There's a rule of thumb, called the Rule of Three, that says if you need to write your own copy constructor, assignment operator, or destructor, you probably need to write all of them.)