How to arrange sabbatical leave?
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.