If by "private messages" you mean private messages between all chatters on the network, then I guess that depends on your network whether it allows the visibility of such private information (I'm not an expert with ircds so I can't tell if there are such ircds).
Then, the answer is simply: If you can log it with any other kind of irc client, then you can also do it with an eggdrop. In essence, eggdrop is just another client, only somewhat more automated compared to other clients such as mIRC.
Eggdrop is not, however, any kind of service or server software, and can't do things otherwize prohibited to clients in general.
Looking through the documentation for IRCD-Hybrid however, I can't find any means to make the server send client-to-client PRIVMSG's to a third party. Without this, eggdrops can't log any such information, just like mirc or any other irc client would be able to.