So people will want to protect their data & privacy, especially companies.
People will want to have a better search experience, which means that things that are on the internet that they want to see will be there. Google is probably the most monitored for takedown requests/deindexing and they control most of the video portals and certainly the window you view the interent through so they are a target to legitimate and illegitimate takedown requests.
edit: they also have built the components and outsourced them. Hadoop/Map Reduce, most of Apache's products, elasticsearch, and another google offering tensorflow can certainly be made into a search engine. Obviously not google scale, but certainly an individual.
If anything, I think people want to have more of the interesting parts of their lives be more public (in other words, they want to be "famous", whether just among their friends or on a bigger stage). They do want many things to be "private" in the sense that they don't want certain people to see certain things online. But I think we're deluding ourselves if we think they care at all about the fact that google is scanning their emails or search history, or that twitter is mining their tweets, or that snapchat is using their location data.