However the code runs on Google AppEngine. This means setting up your own instance takes less than 5 minutes. You don't need your own hosting. You don't have to pay any money for hosting unless you get enough traffic that you go over AppEngine's free quota: http://code.google.com/appengine/docs/quotas.html
In general my advice is that you sign up to both Identi.ca and Jaiku.com (as they're the single biggest instances of both Laconica and JaikuEngine) to get a feel for how the two systems work. Not only will this give you an understanding of how they compare to each other but you'll see how they differ from Twitter and be able to interact with the respective development teams.
Generally Laconica is a little further down the path but both are mature enough for large scale use. Jaiku has a advantage of running on Google App Engine, which means it can be setup fast and scale super large. Laconica has better support in clients because of the Twitter like API it uses. Federation support is also unique to Laconica at the moment but Jaikuengine does have plans for that and the Twitter like API support. The only other thing which separates them is one is Laconica is PHP based and is AGPL while Jaiku is Python and Apache licensed. So as Evan says in the comments
So, wouldn't it be awesome if this were the main question people were deciding right now?