How to arrange sabbatical leave? Sabbatical leave is very common as it is nice to spend a year as a visiting professor in another university and experience a new environment.

It is very beneficial for the host university to have temporary faculty members free of charge (no salary is normally paid by the host university). (1) How do host universities attempt (if they do) to attract visiting professors? Of course, there are job advertisements for hiring visiting professors, but I think they are paid positions and different from normal sabbatical leave.

Visiting for research purposes should be arranged with the leader of the host research group. Thus, the arrangement is at a personal level (somewhat similar to hiring a postdoc researcher.

Visiting for education purposes should be arranged at the level of the department chair.

(2) How can a professor find a visiting professor? Is it chancy? to meet a colleague interested to host? Or s/he must contact many professors and department chairs to find a vacancy?

(1) &amp; (2) Who should actually initiate this process? The guest professor or host university?

(possible 3) Is there a system to facilitate this process, as it is of mutual interest, or everything is left to chance? For example, European Union emphasizes the mobility of students/professors through different programmes. Is there such a system for sabbatical visits (in its classical form of completely working at the host university for a period of time, not guest lecturing as it is common in Europe).
I do not have direct experience with sabbaticals, however the professors I've known that have taken them have been explicitly sought out for the respective sabbatical at a particular institution (as a visiting professor), by colleagues that they personally know professionally (i.e., from being in the same field and interacting via conferences, collaborations, etc.).

In other words, it boils down to networking, and almost certainly doesn't come out of nowhere.  If a professor is planning a sabbatical, he or she will probably already have colleagues willing to host, or will start the process by calling up a colleague and pitching the idea.  I would gather that it isn't likely that there are too many successful cold-calls to departments that lead to sabbaticals.