The problem is that on the Relationship Widget, when the Second Contact has a giving summary that is greater than 0, it will appear when the relationship is selected. Because of this, giving summary permission is required. It is currently not possible to remove that value from the Relationship Widget.
 
In the meantime, a workaround that would allow users to see relationships without seeing giving summary information would be to add the Relationship Related List to the layouts for those user profiles.