So I've just tested a PNG with transparency in the image widget on a brand new install of wordpress 3.5. It worked fine.
I have 2 ideas as to why I'm suddenly seeing all this traffic about PNG problems.
1. In 3.5, I think WordPress updated the graphics engine to support swapping between GD and Image Magick depending on your server. This means that by upgrading you may have unwittingly switched graphics libraries and may be experiencing a side effect of that. If this is the case, this is completely unrelated to the image widget. You would be sing this same problem in any image uploader in wordpress if you use the same png.
2. Coincidence. Perhaps it is really something unrelated. A quick search on the forums for "PNG and White Background" yields a lot of results including many form a year or more ago. http://wordpress.org/search/png+white+background?forums=1. Again, these are all unrelated to the image-widget.
As far as I know, there is nothing that the image widget can do to an image that would cause this outside of bugs cause by WordPress core.
Just to be sure though... Can someone who has been having this problem, please try uploading the exact same PNG that they're having problems with, using the exact same settings (if you're using a resized version, then use the same size), and upload to a post and confirm that you see the same problem in the post?
I'm marking this ticket closed for the time being because I really don't think this is an image-widget bug.
p.s. When I skimmed through the search results, I read a few cases of people using PNG-8 and having white backgrounds where as PNG-24 was fine.