No, dedicated meaning an unmodded dedicated server as downloaded and automatically updated by the official Conan Exiles Dedicated Server Launcher, which is available in another forum post over here:
Conan Exiles Dedicated Server Launcher (Official Version: 1.2.5 / Beta: 1.3.1)
And unless Funcom is maintaining a different version of their dedicated servers they use for official servers (highly unlikely), there shouldn’t be any difference in behavior between public or private servers unless mods are involved or default server settings changed.