QUOTE
Top is usually fatal
Bottom. You can get machines. Might work.
Depends on what you call "top" and "bottom". The side that needs to be clean and scratch free is the side without the label or where you might write on it. It's usually on the bottom when placed in any CD/DVD player/reader I've ever seen, that's usually why it ends up with so many finger prints; the top side gets a thumb but the bottom gets three or four fingers! Of course, one should always pick these disks up by the edges — except when they fall into the peanut butter and jelly sandwich before the two slices of bread are mated...
Since you already have a new disk, why not experiment with the 'damaged' one. These things are basically coated with lacquer, so you'll need to be careful what you use for a solvent. Likewise, the cloth should have fibers as soft as a babies bottom after a good soak in a warm tub! Apparently wiping from the center out or vice versa is the standard advice for wiping. Seems counter-intuitive but who am I to argue? 
Good luck! If you're successful, you might have a new job on some S.F. street corner!
Er, well I beg to differ about the side on which damage is fatal, Jim! From Ryotec's site:
QUOTE
Contrary to popular belief, data on a CD is stored just beneath the label. The mass of a CD is in the polycarbonate plastic that protects the information. Damage to the label side (top) of a disc is unrepairable but doesn't necessarily mean the disc is ruined. Unless a label side scratch permeates the foil layer just under the label of the disc, the damage will not cause the disc to skip. Foil layer damage caused by deep label side scratches or exposure to extreme elements can permanently damage the disc. Fortunately, most damage that causes a disc to skip or freeze is caused by play side damage. This type of damage usually results from normal, repeated handling and play. Scratches, scuffs, marks and even fingerprints can cause a disc to skip or freeze. We can resurface and repair 99.9% of all discs with damage to the play side.
The foil is on the TOP - under the label - but still vulnerable to damage. If you hold a CD or DVD up to the light, you can often spot damage like pinholes or deep scratches. Those cannot be repaired, unfortunately.