If the cameras are recording to a NVR (network video recorder) then the recording will keep going even if the internet is down. You won't have access to it from outside your home, but they will keep recording.
However, if the power goes out, then everything stops UNLESS you have the cameras (and supported hardware such as the NVR) connected to a battery backup.
re: Do Homekit security cameras keep recording if the internet is down or the burglar cuts the power?
I like the features of the Logitech Circle View, I'm looking for a similar one that works with HomeKit but that it keeps recording if the internet goes down, and syncs to iCloud the footage once the internet is back up.
Regarding power, I can solve that with an UPS for example
Look and see if the cameras you're interested in has a battery option (indoor) or solar/battery option (outdoor) vs hardwired power. That will alleviate your power concerns. But you would still need a UPS battery and a NAS drive to save footage.