Both should if your goal is to not have a reusable fingerprint (which for a privacy focus would be). Server should change more frequently since it has access to less information about the browser. Server based fingerprinting is fairly unreliable, client side uses Javascript to generate more bits of unique data.
And despite having cloud integrations, if my WAN is offline I can still view my doorbell. I haven’t tested it HA received notifications or anything though, as I just rely on the Unifi Security app.