It is important to remove this noise signature so that you cannot be tracked down. I cannot guarantee any of these methods will work beyond the shadow of a doubt because the woman doing research for the government on how to find the signature is very good. Check out her papers if you're really good at dense math, and pass along what you learn. I can only promise that this will make their work more difficult.
Step 1: Remove the Camera Info
When you save, I recommend using a JPG compression with a quality setting of no better than 60%. This will add a lot of ugly noise make it slightly harder to create a noise signature.
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Step 2: Clean the Lens
Step 3: Crop and Resize the Image
The way to get around this is by cropping and resizing the image. This will shift the position static pixels and when images are compared together make them just seem like noise.
This alone will not always remove the noise signature itself, but it is one way to make it harder to identify it.
Step 4: About the Noise Signature
The noise signature itself is actually due to thermal noise (environmental heat) and imperfections on the CCD itself. Professional photographers and astrophysicists have tried to compensate for this by creating cameras that evenly lights a CCD a moment before the image is taken to create a noise reference against the image being taken or by super-cooling the camera itself so that heat can't distort the CCD.
In your store-bought camera, it won't have either of these functions and so you will end up getting a lot of pseudo-random noise mainly on the last 3 bits of every pixel. This noise isn't really noticeable to the naked eye, but with a large enough image sample, they can create a noise profile and match almost with certain accuracy pictures from the same camera. Since the noise is dispersed throughout the image itself and tends to remain constant through certain standard image manipulations, it is hard to remove without destroying the image.
Step 5: Removing the Noise Signature
It is probably more effective to remove the noise signature by creating more noise. One thing to try is to open Photoshop and save an image as a jpg using bicubic compression, resize it, crop it and then save it for web, again as a jpg (removing the exif) and finally save it for web again as a jpg using a quality of about 50. This should destroy the noise signature beyond recognition, but the image might not look too good.
It wouldn't be good to do this ever time either since you will end up creating a new "signature." The best thing to do is to save each image using a different elaborate method every time. For instance, you might convert something to a flash video, take a screenshot and then save for web as a jpg compressed with a quality of about 50.
It may not work, but you could also try adding a small amount of random noise to the red and green channels. This might actually clean up the image and remove the signature, but I'm not sure.