Being able to see anyone's bookmarks is a symptom of a privacy issue - that users are not given the choice of what, when, or how to share some parts of their personal information. Several proposed answers ridicule you for bringing this up, or claim that being able to hide your bookmarks has no value. These answers miss the point - that privacy is a complex issue and different aspects of it have different amounts of value to different people.
The problem with making a list of bookmarks public is that a decision is being made to share data without explicitly asking the user whether it should be shared, nor giving them the opportunity to correct it.
Publishing bookmarks is fundamentally different from publishing other pieces of user information such as questions and answers. Questions and answers are created and submitted with the primary intent of sharing information. When I ask a question or write an answer, I am making a choice to publicly post that information. Importantly, I am also taking an explicit action - I type text into a textbox and click 'submit'. As a user I understand that clicking 'submit' will publish the text I have typed for all the world to see.
Contrarily, the bookmarks feature does not have the primary goal of sharing the information that it generates. Users mark questions as bookmarks mainly so they can more easily track them and refer to them later{1} (cf. the faq - "This lets you find it easily when the question may get lost"). Publicly sharing that bookmarking is a side effect of the user's action, and not something they specifically approved when they clicked on the star. And importantly, this side effect is not immediately obvious the way it is for submitting questions and answers - many users do not realize their list of bookmarks is being made public.
The issue here is that users are never given the opportunity to decide or control whether this piece of information - their list of bookmarks - is published. This issue is compounded and confused by the fact that different people who use Stack Exchange sites have different assumptions about what should happen with that information. Some people believe that bookmarks should be public, and so there is no problem. Some believe that bookmarks should be private, and are surprised or concerned that a) they weren't asked or told about the decision to publish their bookmarks, and b) that they cannot take any action to correct that decision for the future. Still other people never think about it, and so we don't know what their preference might be.
If you are in the second group and feel that bookmarks should be private, it may seem like a poor user interface, unfair practise, or just plain deception that the decision to publish that data is being made without your consent. That is a problem, and thus I believe the issue of whether to publish bookmarks should be addressed.
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To further advocate that the issue of whether to publish bookmarks should be addressed, let's look at some popular counter-arguments.
1. "Bookmarks are made public so anyone can see the topics a user finds interesting".
This is certainly a valid and beneficial feature. But shouldn't each person be able to control whether their bookmarks can be used that way? Shouldn't they be given the choice?
2. "Everything you do on the Stack Exchange network is already public"
Maybe it is right now, but does that mean it should be? Several users have commented on or answered this question stating they would like to be able to keep their bookmarks private. The very fact that this question is being asked means the feature has value to some people. Evaluating the possibilities and then making a decision is a better course of action than saying "we can't do that because we've never done it before".
3. "Publishing a user's bookmarking behaviour doesn't reveal that much"
It does reveal something, and that revelation is obviously worth different amounts to different people. Some people don't care if their bookmarks are published. Some probably want them published. And some don't. Shouldn't Stack Exchange accommodate all groups?
4. "You shouldn't post information on the public internet that you don't want people to know"
The issue is that when someone clicks the star to bookmark a question they aren't posting information to the internet. Or at least, they're not consciously choosing to post information, nor asking Stack Exchange to post information. They're mainly just trying to bookmark the question, and a decision to publish information is being made for them. See the explanation above on side effects.
5."Why do you care if your bookmarks are public?"
and
6."Can't you stand a little criticism?"
Behind statements like these is the incorrect argument "if you're not doing anything wrong then you've got nothing to hide". Not only is this not true, it insidiously sidesteps the issue and often leads to heated debates.
To quote Daniel Solove: "The problem with the nothing-to-hide argument is the underlying assumption that privacy is about hiding bad things. .. [it also] myopically views privacy as a form of secrecy". "Privacy" is a complex set of issues. "[It] involves so many things that it is impossible to reduce them all to one simple idea. And we need not do so."
There is at least one important issue here: a decision is being made about what to do with some user information without asking their consent nor giving them the opportunity to change it. The issue is affecting how and whether some people contribute to Stack Exchange. Let's not hide the issue by saying it doesn't matter or criticizing people who discuss it. Let's evaluate the options and solve it. To quote Solove again: "People don't acknowledge certain problems, because those problems don't fit into a particular one-size-fits-all conception of privacy. Regardless of whether we call something a 'privacy' problem, it still remains a problem, and problems shouldn't be ignored."
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Finally, an argument for implementing private bookmark: leaving this problem unsolved hurts Stack Exchange because it discourages some users from contributing.
I would hate for someone to avoid contributing to Stack Overflow because they didn't want their bookmarks made public. Far better to give them the option of sharing what they wish, and then contributing how they wish. We Stack Exchange users want to make the internet a better place. Letting people decide exactly how they assist with that goal can only lead to more positive contributions.
The Stack Exchange Privacy Policy says "We take the private nature of your personal information very seriously, and are committed to protecting it." I'm glad they feel this way. Giving people the choice of whether to make their bookmarks public is a good step towards helping protect personal information.
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Footnote:
{1} Sure, some users may mark bookmark specifically so they can show off what topics interest them, but this is an emergent behaviour.
bookmarks
since, I think one of the regulars should edit the post to update the wording? This post is currently in the review queue lets see if someone edits it.