Maybe, but somehow it didn't matter very much back then. I remember using private chats mostly as an addition to calls, i.e. when I wanted to send someone a link or a file I was talking about. If I wanted to just send a message to someone regardless of whether they were online, Skype wasn't really an option I considered, it was ICQ, later VKontakte, and now Telegram.
Group chats in Skype though, those were popular. Nothing else had good group chats at the time, but then again, after VK introduced them, everyone I know quickly moved there. I don't know how message delivery worked there, but you could receive messages that were sent while you were offline just fine. Maybe you got them from any one online participant, or maybe the "supernodes" did some sort of store-and-forward thing, or maybe a bit of both.