I understood this section to mean that the smart monitor runs its own OS and sends its own network traffic, not depending on software running on the PC:

Smart monitors connect to the Internet and run their own operating system (OS) and apps. They can be useful for performing tasks, like video conferencing, watching a show from a streaming service, or managing your smart home without attaching a computer.

LG’s and Samsung’s smart monitors use the same ad-serving OSes that their respective smart TVs do, meaning they’re poised to use automatic content recognition (ACR). Users of LG and Samsung smart monitors, including reviews site RTINGs, have shown the displays being able to track user activity.

So then you'd have the same challenges as with a smart TV: unless you block its network access, it can report on every image you display. And I've heard that some smart TVs refuse to function unless you give them internet access, so smart monitors might work similarly. Then the only solution would be not to buy a smart monitor, even if you're in full control of the PC.