The most logical way would be to a copy of your vault stored locally when you first login (then kept in sync using shared secret with the server) - as far as I

The most logical way would be to a copy of your vault stored locally when you first login (then kept in sync using shared secret with the server) - as far as I known most password managers work this way (no idea for bitwarden). This should enable schemes when it can check locally if your vault contain a given website, rather than having to reach a server for each website you visit.