People usually make a mistake of performing lossy re-encoding, seeing the file size drop, and concluding it's a success. That's meaningless for a format vs format comparison, because it doesn't control for equivalent quality. You can re-encode any lossy format, even back to itself, and always see significant file size reduction due to nearly-imperceptible quality loss (but that "nearly" matters a lot: see heap paradox).

Proper comparisons need to control for visual quality, but this very hard to do properly, because human range of "looks the same to me" is bigger than compression efficiency difference between WebP and JPEG. So you need either synthetic "objective" metrics that are not exactly human judgement (problematic, because e.g. PSNR likes blur more than people do), or test with lots of images and lots of people.

Anyway, WebP has also clear limitations compared to JPEG. For example 4:2:0 chroma subsampling of VP8 means it automatically loses whenever an image needs sharp colors, because it just can't do it, but JPEG can. VP8 also uses "studio range", so it has less than 8 bits of color precision (224 intensity levels instead of 255).