In Google's case, the company is required under the DMA to share data that would be useful for training a search model.
Why build a European search index?
Currently, alternative search engines like Ecosia, Qwant and DuckDuckGo don't develop their own back-end infrastructure. The new venture will see them build their own search index from scratch, however, amassing results from a mix of different search engines. Ecosia last year switched to a mix of Google and Bing search results.