Depends on the Quality setting and version of jpeg. Even the original jpeg, on high quality, will result in little to no data loss. IIRC, Jpeg can even do lossless, with the only caveat being that it doesn’t save alpha channels (but screenshots don’t need to have transparency, anyway). Newer versions of jpeg, such as jpeg-2000 (and the much less broadly supported jpeg-XL) have much better compression and provide higher image quality at lower file size.
“jpegification” or “Deep-frying” only really occurs with the original jpeg at low quality settings.