Is it possible/ethical for PhD students to collaborate with a professor from different institution on a topic different from their dissertation research without telling their advisors.
Are PhD students limited to exclusively work for their advisors or it is legitimate for them to do research during their spare time to publish papers with other collaborators.
The answer to both the title and the second paragraph of the question is the commonest answer to questions about PhD studies: "Discuss it with your advisor.". The advisor knows about relationships between your department and other institutions, and may know more than you do about the history of collaborations with them. There is a serious issue of whether you can make fast enough progress on your dissertation research if also pursuing other research. The advisor is supposed to give advice about how to proceed with your research career.
The answer to the first paragraph of the question body, about not even telling one's advisors, is that it is just not going to work. As soon as you co-author anything with the other institution, the collaboration is public. Even if your advisor does not come across it, someone will mention it to them. At the best it will be something like "I see your student user2987 is doing interesting work with Professor X."
Maybe the advisor would have liked the collaboration if it had been discussed in advance. If they find out about it indirectly, they will know you have gone behind their back. That is likely to damage your relationship with your advisor.
Is it possible/ethical for PhD students to collaborate with a professor from different institution on a topic different from their dissertation research
Of course it is! The goal of any PhD program is to develop strong independent researchers. Limiting any PhD student's collaborations to one faculty member or group would be directly opposed to that goal. I consider publication of at least one paper without the advisor to be an iron-clad graduation requirement in my field, and in particular for my own PhD students.
However, if your advisor is paying you as a research assistant, then you do have an obligation to work on the specific project for which you are being paid. You can't ethically work on another project—research or otherwise—at the expense of the work you are being paid to do. But as long as you are fulfilling the terms of your funding, what you do with the rest of your time is up to you.
Note: Your advisor may disagree with my opinion about independent work. Working on research without their explicit approval may violate their expectations for you as their PhD student, possibly to the point of them refusing to work with you further. While I believe such a response is both deeply unethical and counterproductive, that doesn't mean it can't happen.
without telling their advisors.
No, no, no. Don't even think about it.
If you feel the need to hide your research activities (the entire point of getting a PhD) from your advisor (the one person whose job is to help you succeed), then your relationship with your advisor is deeply broken. If you lie to your advisor, they will almost certainly find out later. If you can't trust your advisor, find a new advisor. If you think your advisor would disapprove of your independent work, either live with the disappointment or find a new advisor.
