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Your own firearm should not set off your own alarm.
This is the problem that makes firearm detection hard in a house, and the reason most of this category does not sell into one. The answer is arming state and zones, not a better model.
Check your cameras first
This works with RTSP or ONVIF cameras — Reolink, Amcrest, Ubiquiti, and similar. It cannot work with Ring or Nest, which do not expose a stream to anything outside their own apps. Saying so here is cheaper for both of us than finding out after you have signed up.
Arming states
Armed away — nobody should be inside. Every camera alerts.
Armed home — you are in and moving around. Only outside cameras alert, so your own lawful firearm indoors is not an event.
Disarmed — only exterior cameras alert. Everything else records without notifying anyone.
Framed as "an armed person here while we are out" rather than "a gun is visible". It is a different question, and it is the one worth answering in a house.
A person always looks before police are sent
Home dispatch always goes through a monitoring agent who checks the camera first. This is not optional and not a paid upgrade. The nightmare scenario for this segment is armed police arriving at a house where the armed person is the person who lives there, and a human in the loop is the mitigation that actually addresses it.
No face recognition. Ever.
Household members are not enrolled, recognised, or distinguished from anyone else. Arming state and zones do that job without collecting biometrics from your family, and without the legal exposure that collecting them would create.
Runs on hardware you may already have
Ships as a container for a NAS or home server if you already run one, or as a small appliance if you do not. Either way, analysis is local — video does not leave your house, and only the cropped frame attached to a detection is ever uploaded.