Well, based on a few things I've gleaned from Othin and a few other users, as well as the SF page on it...
Double Bonuses increase core stats(STR, DEF, etc.) by a set amount based on the secondary character's(Doublee?) class. Note: Add +3 for any stat they might have capped while they support a unit in double.
Based on their support levels, double units get flat + amounts to hit, avoid, critical, and critical avoid. There are 12 "levels" of Support rank, which can bring the added bonuses for the aforementioned attributes to +20.
The supporting character has a chance to hop in on the main character's attack with an attack of their own. This is a % chance, modified by both characters' support level, and can be further increased with the skill "Dual Attack+".
Along with the dual attack feature, supporting units can also hop in on an incoming enemy attack to mitigate damage for the main. Activation is also dependent on support level, and can be increased further with "Dual Guard+". Note that Dual Guards always negate all damage.
(Credit to Remnant Sage for clarifying)The supporting unit in a double will always gain EXP from an activated Dual Attack as long as the attack connects and hits for more than 0 damage.
Given the +Stat bonuses each unit gives based on class, +SPD and +DEF seem to be the most useful on the harder difficulties to have a supporting character offer in a double. *EDIT: I say this based on feedback from Othin's extensive time playing Lunatic mode. He's claimed that in most cases the +SPD and +DEF bonuses are vital to character survival in most cases because enemies are so fast and tough.* Though I imagine +STR and +MAG are also equally useful.
The supporting character's skills are indeed still active during double, so passive abilities like "Secluded Lady" and "(Crimson) Curse" will still take appropriate effect. This can be quite useful for still giving the main character and nearby allies a boost, even though they're in support and not actively being used! It should also be noted that support characters can activate their own offensive skills(i.e. Astra, Luna, Sol, etc.) on a successful Dual Attack proc.
Those are the main ones I can think of for now. I'll edit or add posts as I can think of more, and I'm sure some of the more experienced folk who've had a chance to play extensively (like Othin) can offer some more insight on the feature. Hope I helped!
EDIT: Thanks to Remnant sage for clarification on Dual Attack EXP.