Step 1: Liquid Nitrogen can freeze alcohol
Geek note: Most drinks are not 200 proof - so the water-alcohol mixtures freeze at a higher temperature than pure alcohol. Still, it's pretty darn cold.
Step 2: Liquid Nitrogen Safety
Since we're freezing alcohol to -114 C, it's important to realize the frozen cocktail itself can be temperature-wise dangerous. At our party, we froze the cocktails, and carefully warned anyone who wanted to try the cocktail-ice directly. In most of the drinks, we also added liquid to some of the cocktail-ice to prevent people from coming into direct contact with the super-cold solids. And in the ones where we didn't, it was probably a bad idea.
I've always gotten my LN2 from a university, or by masquerading as a "remote outpost" of a university to that university's supplier. Renting the dewars isn't too expensive, and the cost of the LN2 itself is less then milk per gallon.