Wanted to correct something. Skype's chat is NOT persistent (if I'm wrong, lemme know.. I'd love to find out there's a feature I'm missing). That is to say, the chat persists.. but only on the distributed hosts that participate in it. So if I sign off at 10pm, and wake up before any of my team members are signed on at 7am.. I won't see anything that happened in between.. until one of them signs on.
This feature alone is probably worth the price of admission to Campfire, IMHO...
Both solutions take about 15-20 minutes to set up and cost nothing, provided you have a VPS to host the server on.
The other way it's handled is by having a bot sitting in channels (doing things like pushing warnings and error notices from your server, too) that records logs. Then you can visit a webpage that the bot hosts with the transcripts on. I'm sure there are packages that perform the logging directly on the server and host them, but it's useful to have some unlogged rooms, too.