This answer is originally from THIS question; I have modified it slightly.
The offspring still has a chance of receiving THEIR hidden ability
Here's the main bit about breeding the Alolan Forms:
In Alola, when breeding Pokémon whose offspring has a regional variant, the offspring's form depends on its parents. If the female parent is the species' normal form (including in a Ditto pairing) and holds an Everstone, and if the offspring is in the same family as that parent, then the offspring will hatch as the normal form. In all other cases, the offspring will always hatch in its Alola Form.
In your scenario, We have two Vulpixes (Vulpix? Vulpi?), one normal variant, and one Alolan. Both have their hidden ability, the normal Vulpix has Drought, and the Alolan variant has Snow Warning. I am assuming this "meeting" is taking place in Alola. We breed them.
If your normie Vulpix is female:
Although both Male and Female can influence the likelihood of a hidden ability on the offspring, Males only affect that chance when breed with ditto, therefore, only the female has an impact.
The offspring has an 80% chance to receive it's hidden ability, Snow Warning.
From Generation VI onward, a female Pokémon has an 80% chance to pass down its Ability slot, regardless of whether it is bred with a male Pokémon or Ditto.
If your normal Vulpix is a dude:
Males CAN influence the ability, but only when breed with ditto, like I've already mentioned. If you breed with your Alolan Vulpix, and she doesn't have her HA, tough luck. With Ditto, the hatchling has a 60% chance to receive it's hidden ability.
Since Generation VI, it is possible for male and genderless Pokémon to pass down their Ability. From Generation VI onward, if a male or genderless Pokémon has its Hidden Ability and is bred with Ditto, it has a 60% chance to pass down its Hidden Ability. Ditto's Ability is always irrelevant to inheritance.
There you have it. When you are breeding Pokemon, you do not pass on the Ability like you would a nature or a move, you pass on the ability to HAVE the Hidden Ability.
So, the Alolan Vulpix offspring has the chance of getting Snow Warning, but NOT Drought.