I think what could be cool is applying this as a personal search engine and marrying it somehow to a personal dns server or squid/proxy server so that you can have a way of harvesting your own browsing data. By using the squid or dnsmasq logs you could spider out urls from it, and build your index automatically.
You can already search your browsing history. Non-centralized personal search engines have a few challenges to solve before they're feasible. 1) The web cannot support thousands or even millions of spiders/crawlers. 2) Search indexes are (probably?) too huge to distribute. See the commoncrawl project. It's TB for a few Billion pages. 3) Assuming a single crawler collects the necessary data, indexes can be easily dist…
Shared browsing search could be a thing maybe as a hobby only though. Probably the only way to make it work is if "you want the privilage to search you must serve too" kind of motto.