Why don't professors give talks at their own departments? Why don't professors give talks at seminars in their own departments?  They travel to other schools to give talks, while seminars at their home departments consists of talks given by visitors.  Is there a reason for this?
In most of the departments that I have known, the professors do indeed give seminars within their own departments. The few exceptions I have known are either fairly small or else have some sort of problem with their seminar culture.

Now, it is certainly true that most talks in any department will generally be given by outsiders---there's a lot of incentives to have visitors give talks (e.g., building collaborations, faculty candidates), and there's just plain lots more researchers outside the department than in it (see Joy's Law).

Still, there will be many possible occasions where an internal talk would be of interest, and I believe that healthy departments see it as a valuable way to keep people in the department connected to one another.