Legal Question in Civil Litigation in Ohio

The owner of the other dog could successfully bring a lawsuit against you if they can show that you (1) owned your dog, (2) your dog caused their dog's injury, and (3) there are damages (such as the cost of medical care).

They would be limited, under Ohio Revised Code 955.28 to the cost of the medical treatment.

They may be able to seek additional damages, however, under Beckett v. Warren, if they can show that you (1) owned or harbored the dog, (2) the dog was vicious, (3) you knew or should have known of the dog's viciousness, and (4) the dog was kept in a negligent manner in spite of its viciousness.

Proving (2) and (3) of the above might be too difficult for them to prove.

In case you were wondering, no, you don't have grounds to sue them. They have grounds to sue you.