That can't be private messaging then. Just clicking some nick can't open window on the destination.
The only thing that I can think of is DCC Chat.
Anyway, See AmIRC Prefs, "Actions", "Special", "Other actions", "Userlist doubleclick:". You can set the doubleclick command here (it can be '/DCC CHAT %u' for example). If you don't need the other side to open up a chat window (just yourself), use '/QUERY %u'.