Some of them do some of them don't. I use a unifi NVR with a local setup which allows me to access it via an app and also via the internet outside of my local network.
However, everything is stored locally, accessible via an app and notifications happen in real-time.
The downside is expense is high, especially up-front compared to a cloud type device.
You can also look at something like a Synology NAS and run their surveillance software with a few cameras (they have huge camera support). Some of them are wifi cams (probably what you want, akin to a cloud type camera), a lot of these though are power over ethernet needing wiring.
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