Earlier quoted context omitted.
> Common Crawl runs once and exposes the data in industry standard formats like WARC for other consumers And what stops companies from using this data for model training? Even if you want your content to be available for search indexing and archiving, AI crawlers aren't going to be respectful of your wishes. Hence the need for restrictive gatekeeping.
Either AI training is fair use or it isn't. If it's fair use then businesses shouldn't get a say in whether the data can be used for it. If it isn't, then the answer to your question is copyright law. Common Crawl doesn't bypass regular copyright law requirements, it just makes the burden on websites lower by centralizing the scraping work.
There is litigation of multiple cases and a judge making a judgement on each one.
Until then, and even after then, publishers can set the terms and enforce those terms using technical means like this.