Airbnb follows its legal obligations to report monies which it disburses to hosts, which will in almost all cases constitute income, via a 1099 (or similar) form delivered to both the host and the IRS.
see: http://www.airbnb.com/help/question/122 , conveniently located near the top of the search results for [airbnb 1099] (if you knew the probable way they were going to report) or [airbnb taxes] if you didn't.
On a side note, it breaks my heart to read "all this money changing hands without being taxed sure seems weird."
Why? Because you don't like taxes on money changing hands? That debate aside, it doesn't matter- taxes are due, they should be paid.
A personal expense does not become a business expense just because a taxpayer calls it that. But states think they can make an interstate commerce tax into a "use tax" just by calling it that.