Matter is just an application layer comms protocol, and yes it's open. Often combined with Thread, a low power radio comms network (The new Ikea stuff uses it), in which case you need a Thread hub, or a Thread border router that acts as a bridge to your LAN and by extension to the Internet. Matter devices may or may not require the cloud.
I have a bunch of thermostats and TRVs (Tado brand) that use Matter, and can be added to my own Matter hub (Home Assistant), to the manufacturer's cloud, or both. There are several brands that follow this model even if they don't use Matter. The good news is that you can continue to use these devices or add new ones even if the manufacturer has gone out of business or shut down their cloud. The bad news is... many of these devices need some configuring; often highly specific settings that can only be done through the manufacturer's app. They'll continue to work but you won't be able to reconfigure them.
Matter allows devices to expose device settings, but it seems that not all manufacturers bother, or they are too specific to fit into the generic Matter types (for instance: heating schedules)