Rather like children - it's always adviseable to stay out of your pet's disagreements - although in this case, with leads involved, it's a bit tricky.
I would agree that you are under no legal obligation to pay, but if you wish to offer a portion of the other owner's bill, then that is a nice gesture, although as advised, you should word your letter so that no admission of guilt is intended, or implied.
From my experience, dogs on leads can get a bit anxious when other dogs get close - maybe they feel a restriction on their 'fight or flight' mechanism, so they get a bite in first.
Hopefully you will be able to sort this out amicably.