Your options will depend on how much effort you are willing to put in and what other services you have access to (or are willing to run).

For example, do you have a Network Video Recorder (NVR) or something like Home Assistant that can consume a Real-Time Messaging Protocol (RTMP) or Real Time Streaming Protocol (RTSP) video feed? Can you modify your network to block all internet traffic to/from the doorbell? Are you comfortable using a closed source, proprietary app to setup the doorbell? Is creating your own doorbell feasible?

I'm not aware of a doorbell that you can buy which meets all of your requirements without at least one of the items I mentioned above. Additionally, I believe the only doorbell that meets all your requirements is building your own doorbell. However, some other brands that will get close to meeting your requirements are Reolink and Amcrest.

I always recommend Amcrest for anything related to cameras. Idk about the doorbell since I don't have that specifically, but the cameras are completely local (no cloud server acting as the relay) and no sign up required unless you use their home app (I use the view pro app to avoid signups).

Reolink doesn't require signups, but their cameras generally require internet from my experience, as they use their own cloud servers as relays, which would mean they can't work offline like you're wanting.

I'm using amcrest cameras with frigate. They work offline, but their doorbell cameras are wireless only, IIRC.

Frigate records and does object detection, so you can get notifications(with photos in the notification) only when there is a certain object in a certain area(like only people in one area, cars and people in another, cats in a third area).

It's a cheap setup if you already have a server running 24/7. It takes quite a bit of setup, but has been trouble free since.