Okay, for Opera I discovered this:
If I create a desktop shortcut and add --private to the end of the target field, it will start Opera in private mode. However, it will not load most of my extensions (most of which are Chrome extensions running through Opera's new addon that allows Opera to run Chrome extensions).
If I start Opera normally, and then click that shortcut, it opens up a 2nd new Opera window, this one private, and with all extensions enabled, since they were all enabled in the first window. The same happens if I select ctrl+shift+n - it pops open a second Opera window in private. For whatever goofy reason, I can't just start Opera in private mode right away because it won't load extensions. I have to have Opera running normally first, then open a new private window. The I can close the other window. Goofy. Is there really no way around this?