First, I'm going to establish a few things, which I will refer back to later on.
- Your friend has to have beaten the game in order for HA Pokémon to start appearing in his/her friend safari.
- Each Pokémon at the Safari has an equal chance of having any of its possible abilities, which means if there's a Pokémon with only one main ability and a hidden ability, the chance it has of having the hidden ability is 50%, and the chance of having the HA is 33% for Pokémon with two main abilities.
- Thanks to an experiment performed by MeloettaMelody, we can assume that the chances of a Pokémon having its hidden ability in a horde battle is approximately 4.8%.
- The probability of finding a shiny Pokémon without the Shiny Charm is 1/4096. With the charm it is 1/1365.3 Source
Ok, so now that we know these numbers, we can do some math to find out the probability of finding a shiny HA Pokémon in any circumstance.
Method 1: The Friend Safari
The Friend Safari is the most reliable of the two methods, and you will find out why shortly.
Pokémon with 1 other ability
Without Charm: 1/2 * 1/4096 = 1/8192 or 0.0122%
With Charm: 1/2 * 1/1365.3 = 1/2730.6 or 0.0366%
Pokémon with 2 other abilities
Without Charm: 1/3 * 1/4096 = 1/12288 or 0.0081%
With Charm: 1/3 * 1/1365.3 = 1/4095.9 or 0.0244%
Method 2: Horde Battling
If you're considering doing horde battles for shiny HA Pokémon, I have one thing to say: don't waste your time. Here's why:
Chances of Encountering HA Shiny Pokémon in Horde Battles
Without Charm: 48/1000 * 1/4096 = 1/85333, which means 1/17067 or 0.0059% when facing 5
With Charm: 48/1000 * 1/1365.3 = 1/28444, which means 1/5689 or 0.0176% when facing 5
So unfortunately, the chance of finding shiny HA Pokémon is miniscule. The most efficient method to use would be to obtain the Shiny Charm and then go to a Friend Safari where the available Pokémon have only one main ability, but even then, only about one in three thousand Pokémon will be shiny with the HA. Good luck, and I hope you are able to find a Shiny HA Pokémon, even if the odds are against you.
P.S. I will be updating this answer as new information comes in regarding shiny HA Pokémon in the Friend Safari (as I am aware that if your friend is online the chances of HAs, and possibly shinies, increases).