No, they don't work on Linux. They're borderline useless. The whole point of client side anti cheat software is to prevent players reading the game's memory or messing with the game's code. There's no practical way an anti cheat can stop someone on Linux because you can just compile a custom kernel that bypasses all the protections.
On Windows you can't do this, so you have to go through one of the known APIs that anti cheat software monitors or find exploits in kernel drivers to get in and poke at the game's address space. They also look for known vulnerable kernel drivers on boot and block loading the game if they find them.
Some anti cheats run on Linux, but they're borderline useless and trivial to bypass.
Unfortunately for anti cheat software to ever work on Linux would require signed and attested kernels and locked down OS software. Something that will never fly in the Linux ecosystem.
Game developers can ship an attested runtime (or hell, even an attested kernel) with the game and, refuse booting it unless the kernel passes some boot tests. Most Linux games already containerize their runtime anyways.
Locking down Linux totally is impossible, but the same very obviously goes for Windows and even macOS as well. Locking down a Linux runtime well enough to play online games seems trivial in my opinion. It's just a lot of work that would be better-spent preventing Windows hackers from pants-on-head insane DMA exploits.
On Windows you can't do this, so you have to go through one of the known APIs that anti cheat software monitors or find exploits in kernel drivers to get in and poke at the game's address space. They also look for known vulnerable kernel drivers on boot and block loading the game if they find them.
Some anti cheats run on Linux, but they're borderline useless and trivial to bypass.
Unfortunately for anti cheat software to ever work on Linux would require signed and attested kernels and locked down OS software. Something that will never fly in the Linux ecosystem.