Okay so after gathering some more information, the three birds aren't like normal roaming legendaries (eg Latios/Latias in Gen 3).
Firstly, as @RavenDreamer mentioned, you must defeat the Elite Four first. Then you can get mewtwo (which is a normal battle).
Then, you must hunt down a legendary bird. It depends on your starter:
- If you chose Chespin, the legendary will be Articuno
- If you chose Fennekin, the legendary will be Zapdos
- If you chose Froake, the legendary will be Moltres
The method of finding a roaming legendary is the same as other generations.
HOWEVER, after finding the legendary, it will immediately flee (you don't get even get a turn).
You must find the legendary 10 more times, after which it will reside in the Sea Spirit's Den, where you can legitimately (i.e it won't flee) fight.