If memory serves me right, the second disk is for the second playable character. So technically, no you don't need the second disk as the one you have has the full game (I'm guessing Dante is the playable character in that one). The second disk is the same game but with Lucia as your played character (with very minor differences in levels).
The disk you are missing is Lucia's disk meaning that you cannot play the game as her. You're not missing out on much because there's only nine missions and a few different bosses and levels from Dante's disk. Lucia handles a little differently and has different weapons/throwing knives but isn't really any more fun than Dante. By the way you got ripped off. Not only did you pay money for DMC2 but you only got half of the frisbees you could of had.
My advice is unlock Trish as fast as possible and then just play the game / bloody palace with her. It's kind of like playing DMC1 but with different (more stupid) enemies.
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