Therefore, sending a query over a public network to a public search engine is not public disclosure. Even if the query was insecure, for a 3rd party to access it would require some sort of sneaky behavior to get the query. Security is a different question. You may be concerned about the interception of messages in transit, or the security of servers to hacking. In either case, the issue is theft, not public disclosure.
All search engines make proclamations of confidentiality.
This should be sufficient to avoid the requirements of public disclosure even in the unlikely case that information was stolen. Also, I am not aware of an actual example in the 25 years or so of electronic databases of theft or disclosure of information sent in confidentiality to a search engine, or of any lawsuits on this same subject.