Transformers aren't zettabyte sized archives with a smart searching algo, running around the web stuffing everything they can into their datacenter sized storage. They are typically a few dozen GB in size, if that. They don't copy data, they move vectors in a high dimensional space based on data.
Sometimes (note: sometimes) they can recreate copyrighted work, never perfectly, but close enough to raise alarm and in a way that a court would rule as violation of copyright. Thankfully though we have a simple fix for this developed over the 30 years of people sharing content on the internet: automatic copyright filters.