IMPORTANT! You will need to register people as Friends on your 3DS/2DS for this. Also note that this requires a Wi-Fi Internet connection for your 3DS/2DS on both sides: it won't work until the Friendship is confirmed.
The Safari Zone returns, but now with a forced-online mechanic and it's called the "Friend Safari". It is not found until beating the Elite Four and visiting Kiloude City - it will be in the northeast corner of town. There, you can choose to go into a particular Safari Zone.
However, this Safari Zone is not based on anything in-game - rather, it is based on Friend Codes. Accessible from the Home Menu in the orangish square, you can register Friend Codes through infrared communication or the Internet through the actual exchange of the numbers. (You get a Friend Code upon accessing the Internet.)
Once you and your friend have each other registered on your systems, and the 3DS's/2DS's have connected to the Internet to confirm this, you should be allowed to enter a Friend Safari.
This Friend Safari will contain two or three Pokémon of a certain type, ranging from Normal-type to Fairy-type. To unlock the third Pokemon in a safari, the other person must have beaten the Elite Four and be online in-game at the same time as you .
Safari Pokémon are special in that they are set with a fairly flat 33% encounter rate, and each Friend Code almost always provides a unique trio of Pokémon. These Pokémon are also very special in that, for one, they are more likely to have the "Hidden" abilities, which are rare and valuable, like a Moltres with Flash Fire (not found in the Safari though!).
Additionally, all of the Pokémon seem to have two IVs equal to 31 - that means, in at least two stats, they will gain 31 points over their base values at Level 100! And, finally, the Pokémon typically fail to obey version-exclusivity, typically are not native to Kalos ... and can even be starter Pokémon! How awesome is that?
It gets better (for some). Firstly, unlike normal Safari Zones, you pay nothing and spend as long as you want (step- or time-wise) in there. Battles in the Friend Safari are that: battles; no more of this "stone, mud, Safari Ball" crap. Pokémon are found at exactly Level 30, and you battle them with normal Pokémon.
Recommended capture tactics include a Quick Ball (or the more situational special balls) on the first turn. If that fails - it normally does - you can use the typical "False Swipe, Sleep/Paralysis, Pokéball" formula. Whatever the case, it makes capture - and thusly Pokédex completion - much easier.