First, each veteran under this program must qualify by either serving during certain dates (i.e., Vietnam, Korea, WWII, etc.) or possess an approved campaign medal, been a resident for at least 3 years.
In this case if both veterans qualify, then who gets the benefit? The owner on title as of January 1st would receive the benefit as they are also liable for the tax as well.
The only way both veterans could claim against a single property in the same tax year is if they both were listed on the title as of January 1st and the property was the primary residence for both veterans.