Below is the reproduction steps:
login as site admin user
1. create a page on a site
2. put bookmarks porlet.
3. click the configuration icon and move to scope tab
4. set Scope as selected Page and Scope page as Current page and save.
5. add some bookmark on it
login as site member user
6. move to this page
Then you can see the one bookmark, but it is just a text and you cannot click this bookmark.