The first part of this paper focuses on competition between search engines that match user queries with webpages. User welfare, as measured by click-through rates on top-ranked pages, increases when network effects attract more users and generate economies of scale in data aggregation. However, network effects trigger welfare concerns when a search engine reaches a dominant market position. The EU Digital Markets Act (DMA) imposes asymmetric data sharing obligations on very large search engines to facilitate competition from smaller competitors. We conclude from the available empirical literature on search-engine efficiency that asymmetric data sharing may increase competition but may also reduce scale and user welfare, depending on the slope of the search-data learning curve. We propose policy recommendations to reduce tension between competition and welfare, including (a) symmetric data sharing between all search engines irrespective of size, and (b) facilitate user real-time search history and profile-data portability to competing search engines.