Puncture wounds are THE most likely to get infected of all the different types of wounds. They are also extremely painful, especially facial and head wounds. I don't mean to sound hard-hearted to your monetary problems but you have GOT to find a way to pay for a vet visit for him. The chances of him having big problems from this if it is left untreated are too great. Either apply for Care Credit at your vet's office or find something to sell or pawn at a pawn shop or take out a payday loan but find a way to pay for his treatment. This is nothing to mess with and certainly not something you can treat at home.
Ghilly
The longer that wound is without care the greater chances of infection and long term problems healing. Some vets accept something called care credit which gives you time to pay off the balance before interest is added. It's essentially a line of credit good at vets, dentists, and other places. They will give you an approval in a matter of minutes in most cases. I hope you are able to get your dog the care he needs.
You can try cleaning it the best you can and flushing it but I would really worry about infection and would not try to stitch that yourself. Agree, see if you can't come up with some money,me vein if you need to borrow it, and have the Vet check it. Better to try now or you might have twice the problems and cost later on.
Always a good idea, if possible, to try and keep a few bucks in an account for the animals.
Hope all works out.
Do you actually think you can stitch up a dog without anesthesia and have him hold still, plus clean the wound ahead of time, plus give antibiotics to prevent infection? My first concern would be to make sure the other dog had his rabies shots, and yours too, then I would find some way to get the money for a vet, whether you have to borrow it from a friend or a bank. Your dog has to be in a lot of pain, please try to find a way for medical help.