Do I need a serial number to reinstall Mac OS X?
Mac OS X doesn’t have (nor had in the past) Serial Number or “Legit Purchase” mechanism. You can buy/borrow/lend/steal/copy a copy of Snow Leopard (or any other edition) and install it on any number of Machines.

Technically this is possible, however, according to the EULA and all the legal stuff, you can’t; in any case, the operating system doesn’t have any copy-protection mechanism.

EDIT: Bear in mind that a plain OS X Snow Leopard disk is not the same as a recovery disk. Although a recovery disk contains the OS, it is usually tied to a particular model, so if you try to recover a Macbook Pro with the iMac recovery disk, that might not work. (This has happened to me in the past trying to recover the other way around, a friend’s iMac’s HDD failed and he replaced it but couldn’t find his recovery disk. We tried to use my Macbook Pro’s disk but it wouldn’t work. The same with my Mac Pro. 

Going to an Apple Store will possibly help in that case. In the end I found my Snow Leopard’s original disk and we used that to install his iMac.