IIRC, this indicates that it's linked to someone else's account, and has not been shared with you. The "beep when moved" feature is to alert people they're being tracked.
For example, I let my mother in law use luggage of mine with an airtag still in it and every time she moved it after the first day or so, it would play a noise.
Rolling your own is probably your best bet as being able to check remotely typically means your camera streams are running through a company's servers.
Personally, I moved from BlueIris and I've been incredibly happy with Frigate which is configured to pull from my local IP cameras - none of which have direct access to the internet:
https://github.com/blakeblackshear/frigate
You can check in via browser assuming you have some method of accessing your local network. Paired with a google coral device (or a beefy CPU) alongside local storage, it also has the ability to detect objects in real-time which often is a feature that requires video processing on a company's server.
For example, I let my mother in law use luggage of mine with an airtag still in it and every time she moved it after the first day or so, it would play a noise.