Tl;dr : yes they are limited in their access to the rest of your PC.. mostly
From what I understand, when such anticheats are configured for Linux, they're still running in the user space and is why some developers go as far to disable support for Linux entirely.
You, the privileged user, unless logged into the root user (not recommended), are part of the "sudoers" group, which allows you to execute commands on behalf of the root user using the "sudo" command which requires your password. Games should never need this to play.
This however doesn't mean the AC is sandboxed, its honestly beyond my knowledge exactly what it does have access to, but I can say it is far less than what Windows kernel AC has. And again why developers feeling the need for such intrusion simply pull away from linux