Deleting files after completing Migration Assistant

Hello! I just successfully completed the migration from an old mac to a new one. But it seems like both user accounts are still coupled (i.e., when I try to delete files from my old mac, it automatically does the same action on the new one). How can I stop this? I would prefer to delete all files from my old mac before signing out of iTunes/iCloud on the old one... but maybe that logic is what's wrong in the first place? Thanks in advance!