JPEG can make some screenshots smaller, especially when they contain large photo areas. For screenshots with tiny text, flat colors or transparency, first check whether the destination accepts PNG. Keep the PNG as your editable original.
Prepare the file
Open the image in an editor and export a JPEG copy at high quality. Choose a background color before exporting if the PNG has transparent areas. Keep the original dimensions for the first export so you can compare the effect of compression by itself.
Check the finished copy
Open the JPEG and compare it with the original at the same zoom. Look around text and sharp edges for blurry outlines. If the file is still too big, crop empty margins or try a small reduction in quality, then check readability again before submitting.