Regarding that, seems like it's actually not that hard to gain the effects, a few of them at least.
Not only do the trainers retain the type between games, but the same Pokemon appear in the Friend Safari for that person, such as my set type is Dark and my Pokemon that will appear are Nuzleaf, Crawdaunt, and InkayJust got to Friend Safari, and since I've got several Staff on my FC list, who appear to be on also on Serebii's own. As far as I can tell you as a Friend share the keep the same type on different 3DSes. Hydrohs is Ice type, on both Mine and Serebii Games. There are other people's names I recognise, and they share the same types.
Gogoat learns Double-Edge at lv 40, Horn Leech at lv 47, Leaf Blade at lv 55 (and I found this when I already PP Upped the Horn Leech move ._.) and Milk Drink (GOAT MILK FOR EVERYONE) at lv 57 (not 58)