I have gone back and forth with what type of search to use. I used to use the Joomla search, then I switched to the Google search. Steve Burge notes that there are some reasons not to use site search.
I might actually go back to using the Joomla search. Not for any of the reasons Steve gives, but more because of a feature of Joomla I recently re-discovered. The search component in Joomla allows you to log the search terms people are using. This is really useful as it allows you see what terms people are using to get to your content. Useful intelligence to make your site more accessible!