Is it possible to do this? If I find a research group/researcher working on something I like and I just want to do my thesis there. Is it necessary that
I suppose the only way to do what I am asking about it via an exchange program,but is there any other way?
This depends strongly upon what university you're attending now, and where you want to do your research. For instance, in Germany, students are allowed to do a master's thesis anywhere in the world, but they are responsible for organizing this themselves. Other schools and departments in other countries may have more restrictive policies.
An exchange program may or may not be necessary, but this is entirely dependent on the specific policies of both the school you're attending and the school or institute you'd like to do your thesis work at. The best guidance I can give you is to first ask the graduate students' office or "graduate officer" of your current department to ask what the relevant regulations are for your home institute. Then talk to the other institute. Of course, start such a process well in advance of when you want to do the thesis work; a semester at the minimum, a year is more strongly recommended, as it may take several months to organize the required paperwork and visas (if you're planning on traveling abroad to do the thesis).
I am writing my masters thesis as an exchange student at another university but it is not necessary to enrol as a student at the other university (like aeismail mentioned). If you need a scholarship then you might want to look for a possibility as an exchange student but it has also additional advantages to go as an exchange student. You get informed by the student administration of the university about several topics.
The most important thing is to ask first at your study administration/students office and afterward to find a supervisor for discussing with him the next steps. The thesis has to be compatible with your home universities restrictions.
You can then search for a compatible thesis and research topic. It depends on the university how they publish the open postions, research projects and theses topics. Most research groups have their own homepage where they publish open positions or topics for a research project/thesis. You can also write them a polite email to ask for an project you are interested in.
There are also some independent institutes or even companies that provide writing a Masters or even PhD thesis (dependent on your field of research) on behalf of them.
