What's a good way to sharpen a carpenter's pencil, without buying a special sharpener?
Whatever you have handy. There are many tools that would work, besides a pencil sharpener dedicated to this.


A pocket or utility knife - cut away from you of course.
A sliding, compound miter saw (overkill, but it will work well enough.)
A belt sander
A band saw
A drum sander in your drill press (or an oscillating drum sander if you have one in the shop.)


The point is, a carpenter's pencil is mainly wood, and can be shaped by any tools you have in reach that can also shape wood.