You should move your sites to the same server and same DB and then you will be able to run whatever queries you. Barring that, the only alternative I know to writing your own search engine is to use google’s site search API which gives you access to the raw data instead of the snippets google usually displays. Cheapest tier is 100$/yr which is probably why you will not find many plugins for it.
Another option is to have some replication of the DBs into yet another server (or the one where the site with search functionality is), and run searches on it. But even that way it will be better if all servers are at least in a local network to have less potential security issues.
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