I highly doubt the use of mods on a private server would result in any infractions even after the anti-cheat is introduced, especially since the majority of the mods available are just normal game-enhancing mods and not trolling programs (Though there is always the potential for accidental action, like workshop content available from steam can result in you getting a ban on dying light because of their system, even some of the content released from the developers themselves as dlc had that issue in that game specifically). I'm not entirely sure what you mean when you ask if it is safe to use them on a private dedicated server, as in you don't want to get banned or will it mess with the files or server itself? I don't believe Endnight plans to issue something like bans for mod users, they seem to be fairly accepting of mods, but are introducing an anticheat to prevent issues like the greifing mod being used so commonly against people who would rather have no involvement with mods. ModAPI is, to some extent, supported by the developers (not as in they develop it, but as in they are aware of it and even have a forum in here dedicated to ModAPI related bug reports and ensuring continued compatability). I don't personally use the mods but a few friends do and so far they haven't had malware issues as a result of downloading mods from ModAPI.

I'd just be careful about clicking any ads or unintended links while using it.

ModAPI is the source for most who get the mods for forest so I imagine it is relatively safe despite the malware flags.

The forest install mods without modapi