Both to some degree. The processor manages each program that runs and helps it with computing the event. The SSD needs to be used to hold the process in memory, especially if you go over the RAM that is built in, on the machine. but also the Finder uses temporary space to ensure the copies go over uncorrupted. If your internal hard drive is too full for any given file, it could seriously delay the copying of the files as the machine tries to better manage the copy. The risk is higher for corrupt copies in that event, and the system may kernel panic, or have a spinning color wheel cursor if it can't find anywhere to move the files.
One of the things that really shows this is when iCloud and Time Machine backups happen. A temporary folder is created at the root while the copying happens for each. And if something pauses the copy, you'll see the folders remain there until the copy is allowed to continue, or you verify the destination already has the files needed.
Always keep at least two copies of all files you need at all times.