The website for this is CouchSurfing.
The website touts that this is very safe, and just like with Ebay, couch surfing hosts are verified and rated by others who have used their couches. This can be a very easy way to save money. Also, it's not always a couch that you get, most people have extra rooms or beds for you to use.
I on the other hand have a hard time trusting someone. I don't like the thought of leaving my stuff at someone's house while I am sightseeing.
What do you think? What would you do to save money? Would you sleep on someone else's couch/floor/room without knowing them? Would it be more about the experience of meeting new people who are locals or about saving money?