In modern systems none of the memory is actually copied just because a fork system call is used. It is all marked read only in the page table such that on first attempt to write a trap into kernel code will happen. Only once the first process attempt to write will the copying happen.
This is known as copy-on-write.
However it may be necessary to keep track of committed address space as well. If no memory or swap is available at the time the kernel has to copy a page, it has to kill some process to free memory. This is not always desirable, so it is possible to keep track of how much memory the kernel has committed to.
If the kernel would commit to more than the available memory + swap, it can give an error code on attempt to call fork. If enough is available the kernel will commit to the full virtual size of the parent for both processes after the fork.