Arcanist probably only has access to Carbuncle, and will probably have it's own set of spells, or "spell summons" that aren't pets.
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I don't really think Summoner will get the big primals. They made them quite powerful and really larger than life in FF14 compared to FF11. If anything I believe Ifrit, Titan, Garuda, Shiva, Ramuh, Leviathan and Odin will be, as they have stated before, summon-able by the Free Companies and only one of them will exist.
Since we don't know the mechanics of how Arcanist/Summoner will work it is really only speculation but they have already confirmed the whole Free Company summon situation. If anything I believe Summoner will get lesser primals.
Some examples are:
Carbuncle
Siren
Cait Sith
Unicorn
Midgardsormer
Moomba
Quezacotl
Phoenix
Typhon
etc
Just look here at all the summons from past games: http://finalfantasy.wikia.com/wiki/List_of_Summons
I'm much more excited to see them do something different and add lots of variety. But then this is just my favorite idea of what they could do.
It doesn't say WE as summoner's can.
"The beast tribes of Eorzea worship and summon forth beings known as primals, among which are Ifrit, Garuda, and Titan."
Says the beast tribes can. If we can summon them then it wont be to the same effect as it was in FF11 I imagine. Maybe only Summoners can summon the one time only primals FOR the Free Companies.
Arcanist may be the new BST job since they can call a familiar into battle.