What is the PhD application process for Germany and Switzerland?

I can speak for Austria (place where I did my PhD) and Switzerland (current place of employment), but my answer should be applicable for Germany as well.

I wonder what is the Phd application process for Germany and Switzerland. Are there scholarships offered? Is there a need for scholarship at all, or do you get paid as an researched directly from the institution where you get the Phd postion?

Both models exist, but the far more common one is that you are just employed as a researcher while doing your PhD. This has advantages and disadvantages. Typically these positions are financially rather attractive (at least as far as PhD student salaries go), currently at about 2400 EUR 14 times a year in Austria for full-time employment, a bit more in Germany, significantly more (around 5000 CHF 12 times a year) in Switzerland. All of these salaries allow you to conveniently live (no need for a diet of Ramen noodles) in the respective places. On the other hand, as a researcher you are not "just" working on your thesis. In addition, teaching has to be handled, support in administrative matters will be part of your job, and you will need to work on matters of your research project that you find neither interesting nor advance your thesis work in any way.

The application process is generally as for any other job - find out what jobs are offered, contact the person per mail, send in CV, (usually) do an interview via Skype, wait for an offer. The hardest part is probably finding out what research groups currently have job openings. Positions are generally not announced very widely (if at all). However, just because you cannot find an ad for an open position with a specific professor does not necessarily mean that he does not have a position available. Both places that I worked at had open positions almost at all times for a really qualified student. What really helps here are connections - do you know somebody who is already an "insider" in academic circles, maybe a PhD student or postdoc? If so, ask him to put you in contact with some faculty.

If not, look for universities that you might consider joining, find out from their web page what faculty there handles your topic, and send them a short mail. Keep it crisp - both of my professors so far have been insanely busy, and any mail from an unknown person with more than one short paragraph will never be read carefully. Just tell who you are, what your current university is, and that you would like to talk about the possibility of doing a PhD. Give them a week to answer and then send a quick and friendly reminder (my current prof is the dean of the faculty, and given the size of his inbox mails sometimes do get lost - that does not mean that he is necessarily not interested). Maybe, the professor will put you in contact with one of this postdocs or PhD students to "chat" via Skype or in person (if possible) - consider this the technical interview, because afterwards the prof will ask the person that you talked to whether they think that you have the technical skills that are required for the selected field. What you should not do is send an overly long formal application - most professors get many of those from rather dubios applicants from the far east, hence formal blind applications are generally discarded unread. Don't be one of those.

is there a need to search and find for a supervisor (or professor), and explain him the idea for the research that you are planning to work on... or do you get the position and work on the topics which are treated in that research group. my question is both for Swiss and DE?

You are generally not expected to "come" with your own topic. Your broad overall theme will be defined by whatever project / position pays your salary, and on top of that you are expected to define your concrete research project together with a postdoc and/or the professor some time into your PhD (say, about a year after starting). What you should know is roughly what interests you research-wise and contact only professors that really do this research. This sounds like a given, but I work in services and software engineering, and I have lost track of many applicants interested in robotics, AI, formal methods, etc. I have already discarded. Don't be one of those, either.

A good place to look for open positions in academia in CH is the ETH-gethired website. In Switzerland, you will usually be hired by the university or research institute and receive a salary (about 3.6 to 4.5k CHF/month). In exchange you typically have to do TA work (assisting with practical sessions in courses, correcting assignments, etc.) and sometimes technical tasks (taking care of lab equipment). In some universities (like the two Swiss Federal Institutes of Technology, ETH and EPFL) and for some programs, a formal application to a competitive graduate school is also required.

In some cases, you can get funded directly by the Swiss National Science Fund which leads typically to a lower pay but less or no teaching tasks. But it is seldom the grad student's task to secure funding.

Scout the lab websites for positions not listed in my link, and don't be afraid to apply spontaneously as well, if a place sounds particularly interesting to you.

In general, doctoral studies in both Germany and Switzerland are paid research employees. In Germany, for instance, Doktoranden (doctoral students) are formally called Wissenschaftliche Mitarbeiter ["scientific (or academic) workers"]. They are paid as government employees according to a fixed scale, and have a contract outlining their duties.

Now, for these programs, you apply as if you were applying to a job outside of academia, except you send a CV in place of a résumé. The hiring is done directly by the group of the professor who has the open position.

There are some exceptions, primarily related to American-style doctoral programs. For these, your application is in a style similar to that of a graduate school in the US (form, letters of recommendation, statement of purpose, etc.). These are typically fellowship-based positions that carry a stipend. They usually also have a reduction in teaching and supervisory duties.

In any case, however, you should never have to pay (or get an outside scholarship) to do doctoral studies in a scientific field in Germany or Switzerland.