There are two distinct reasons why a disk encryption might fill the storage area with random data.
If you're encrypting in place, or more generally if there's a chance that the storage area contains plaintext data that you don't want to expose, then the existing data needs to be wiped, otherwise traces of old data may remain in space that hasn't been overwritten by encrypted data yet.
Wiping with zeroes is enough against all attacks that don't use specialized hardware such as an electron microscope. Wiping with zeroes is enough even against all published attacks with specialized hardware for most storage media. In the 1990s, Peter Gutmann demonstrated that data could be recovered to some extent after overwriting it with zeroes, due to magnetic remanence. The effect is rather small and each bit has an independent chance of not being recovered correctly, so it's difficult to recover data, but theoretically possible, leading to the recommendation of random overwrites.
In any case, the remanence effect is so small on contemporary hard disks that overwriting with zeroes is good enough.. For EEPROM, but not other flash technologies, I've seen recommendations to overwrite with multiple passes of random data. For NAND or NOR flash, overwriting with zeroes is good enough.
To put it simply, overwriting with random data instead of zeroes is useless paranoia in almost all cases where it's used.
There's a different reason to fill the disk space with random data, which is if you consider the size of the encrypted data to be confidential. Encrypted data is indistinguishable from random data if you don't have the key. Therefore, in a disk filled with random-looking data, it's impossible to tell which bits are random and which are encrypted. This is only a very small privacy concern since revealing how much disk space is free is rarely a concern.
If you want to start storing data before wiping, it's more complicated: the wiping tool and the disk encryption tool have to know which area they can use and which area is the domain of the other guy. Also, if you were relying on the free space being wiped or randomized, then your disk does not meet your security requirements until the free space has been properly disposed, so you shouldn't be using it.
In summary, overwriting with random data is rarely useful. But if you do want it, you need to wait until it's done.