Although there are medical uses for silicone, injection into the lips is not recommended and is not approved by the FDA.
You will however, find dermatologists that are willing to do this. A serious and very possible complication is your body reacting to the silicone and creating severe scar tissue known as granulomas. If this should occur, the drops of silicone cannot be removed and a portion of your lip may have to be removed to fix this problem. We have had to perform this in our practice as well.
One reason you may see disagreement in the answers to this question are that dermatologists are usually the practitioners who inject the silicone but the plastic surgeons are the ones who see and treat these patients when they develop the serious complications