I have the enewletter subscription widget set with the "Show Groups" box unchecked, and automatically have new subscribers subscribed to my general list. Only that group is listed as "public". I just noticed, however, that if someone is logged in, they see all of the groups in the widget. I'll attach some screenshots.
My theme will probably allow me to avoid this and remove the widget altogether if someone is already logged in (because they don't need to subscribe to newsletters if they are members), but it seems odd that even though the widget says to not show groups, they all show up when logged in.
I also use the Membership plugin if that has any bearing on this.