Summoner Info

[Xilrasis]

At 30, I was able to do both the SCH and SMN quests as ACN, with only CNJ/THM above 15 for other jobs (Started today with CNJ 20, THM 20, ACN 20, PUG 12, ARC 4).

SCH pulls abilities from THM and CNJ.

SMN pulls abilities from THM and ARC. (they DO NOT have access to aero, as a result)

The quests said nothing about requiring any particular classes at any level other than the obvious level 30 prerequisite for Arcanist, so draw conclusions as you will... but since I had both DoM jobs at 20, there isn't really anything conclusive to draw.

Both quests began in the ACN guild after you complete the level 30 class quest, and SCH sends you to the Marauder guild (Where your scholar contact will remain for later quests), and then subsequently elsewhere to complete 1 Duty quest.
Your contact for SMN is in Gridania, and is followed up similarly with a Duty quest.

*EDIT*

I suppose that the definitive piece of information you can pull from this is that there are at most 1 required classes for each job, and the second is definitely secondary simply for abilities.