Bitwarden currently doesn’t have a concept of data conflicts or versioning of an entry as a whole (i.e., not just password history). So for example, if you make a change in the browser extension (say, change the password) and then make another change in the mobile app (say, add a URL), then your password change would be lost because the mobile app (when used with bitwarden_rs) wouldn’t have gotten a notification to reload the updated entry. On the other hand, if you made the change in the mobile app later on, after it had done a sync, then there wouldn’t be a problem.
These data conflict/overwrite issues are largely mitigated in the official Bitwarden with the combination of WebSocket and mobile push notifications, but as you noted, bitwarden_rs doesn’t currently support mobile push notifications. It might be possible to add mobile push support for people who have purchased a license from Bitwarden (and thus have a license key), but I’m hypothesizing on this point, and I’m not aware that anyone has investigated the feasibility of this in detail.
Visiting https://vault.example.net/notifications/hub with a web browser is just going to give you Unable to parse WebSocket key, since a web browser won’t attempt to negotiate a WebSocket connection without some JavaScript code doing that specifically.
I usually test WebSocket notifications by just opening two WebSocket-enabled clients (web vault, browser extensions, or desktop app) and pulling up a secure note entry (other types of entries work too) in both clients. If WebSocket notifications are working, when you edit and save changes in one client, the changes should be immediately reflected in the other.