Distinguish where? The icon can be the same, and on Bookmarks page could have topbar tabs for Private and Public.
Discussion
I might want to use a different icon, or maybe add a lock icon.
Not necessary. Doesn't make sense. Bookmark is a bookmark, doesn't matter if is private or public. And doesn't make sense have the same note being bookmarked public and private at the same time for example.
If someone wants to know if it was bookmarked is public or private, just click on the icon and will show if is Private or Public, with option to uncheck or switch between public and private.