If you are seeing puss, it's already infected.
It's a gamble whether it will heal on it's own, and unfortunately, the odds aren't that great. (a bit better than with cats, though).
If it doesn't, it could abcess, and require expensive treatment.
How deep and long is the wound? If it's fairly superficial, (not too deep) you could try using a topical antibiotic like Neosporin. Apply per directions, and hope your dog doesn't lick it off. Reapply as often as it's removed, and watch carefully for signs of worsening infection.
If it's more than about 1/8" deep, open, and oozing puss, it's time to go to the vet for antibiotics.
Generally, treating an infection earlier is always better than later, and can cost much less. Often, all that's required is a shot of antibiotic, and a course of pills/liquid to take home. This isn't too expensive.
If the wound abcesses, it may require surgery to clean it up, and that gets expensive. (general anesthesia, sutures, etc.)
Many, many years ago I worked for a vet. We almost always saw cats and dogs after the infection got to the point it was abcessed. Not pleasant to to clean up the wound and dead tissue, etc. Recovery time was much longer than if the wound was treated early, too.
Good luck! (maybe put something on the bedframe that cut the dog; even duct tape would help round out a sharp edge.)