Gold coins carry no built in tracking system, so no one can trace them the way authorities trace bank transfers or registered securities.
However, some coins do leave a paper trail. Dealers often record buyer information for large cash purchases to comply with reporting laws, and many countries require dealers to report transactions above a certain threshold.
If you buy through a dealer with a credit card, check, or wire transfer, that transaction creates a record. Coins with unique serial numbers or certifications from grading services can also be identified if someone documents their ownership.
Still, physical cash purchases of common bullion coins remain largely untraceable, which is part of why people value gold for privacy and portability.