Obviously, hosting a search index locally would be best practice, for the search part, but usually comes with quite a significant hardware cost and operational complexity, if you want to have an up-to-date index. I'm not aware of good self-hostable options in the space but would be happy to hear about them if anyone here has experience with that! AFAIK LinkUp and some other provides do offer private cloud/on-premise deployments, but I think mostly for their biggest customers.

For the web fetch/access part of the equation, that is inherently live and not containable for a local deployment (unless you would host a whole internet archive/Wayback Machine, which is even more of an undertaking). There, the main options you have to prevent leaks, are a domain allowlist (if feasible), doing query parameter stripping (error prone, and still no 100% guarantee), or configuring the client (e.g. our tool) in a way that users have to explicitly allow those tool calls, and are informed about the posed risks. We are also working on a system, where certain tools are fully deactivated once e.g. an office document that's marked as confidential enters the context.

For the highest requirements on company secrets, an air-gapped system that doesn't have web search/access capabilities is obviously the ultimate solution.