A friend of mine has discovered over 100 asteroids, mostly while living in Mexico. I've wondered the same thing. I'll have to ask him...
EDIT
Here is his answer:
"The short answer is, they are tracked in computer data bases - easy to identify duplicates. The longer answer is, the orbital elements of each asteroid are as good as fingerprints for identifying them. For an asteroid to be numbered, the orbital elements must be precise enough to permit prediction of its position within 2 arcsec after 10 years. The key to precise orbital elements is a long observed arc."