Uh, no.
.PNG saves the image exactly as it is and doesn't go through compression when saved. This is why most people save their work as .png, despite large file sizes.
.gif saves everything in 256k colors. So, you might lose some details in color, but any major loss in quality is done by compression, you can get rid of this by saving and preserving exact colors at 100%
.jpg images go through compression when saved to make them smaller. This is why if you save it once and host it you won't see any loss in quality, but after it's hosted and saved several times by different peers you will see a loss in quality.