Is it realistic for a postdoc to negotiate for a private office?
The big issue in having a private office is available space.

Most departments have a limited amount of space available, and what is given to any particular research group is correspondingly restricted. There may not be a private office available to the group, in which case it wouldn't be possible to accommodate such a request. In addition, if there's a more senior person, then usually such an individual would have a "higher claim," so you might not get one.

Of course, if you never ask, you'll never know.