How do the divisions work?

Each division is focused on specific actions. Completing these actions benefits you in-game. Every day there is a tally of division activity, and the most active divisions are rewarded. Furthermore, just belonging to a division supposedly endows you with special benefits based around that division.

How exactly do these mechanics work? I joined the Interceptors, but I've gained bonuses for other divisions (the achievements, and had the bonuses appear on my screen when they activate). What impact do divisions, and your choice thereof, have on offline play?

Solution 1:

Divisions are essentially your currently chosen "group". I haven't actually played the game (yet), but I'm reading through the strategy guide, and here's what I'm gleaning:

When you're playing in a squad, you have a chance of benefitting from some powerful buffs if there are other players hanging out in the same zone. This is division support in action. The same way that the division you belong to grants you a buff, so too do other players' divisions, and if you're in their vicinity, you reap the benefits. And if you happen upon a BLADE on the field, you can request their division buff instead of scouting them.

From what I can tell, at a basic level, you will want to choose a division that gives you more or less a permanent buff while playing. Running into others is a bonus, and you can get another buff from them, if you're willing to forego scouting them. There's even a table for the buffs!

Division          Buff Name             Buff Effect
---------------------------------------------------------------------------------------------
Pathfinders       HP Support            Recovers HP over time.
Interceptors      Ranged Master         Increases the damage dealt by ranged weapons.
Harriers          Melee Master          Increases the damage dealt by melee weapons.
Reclaimers        Drop Sensor           Increases the chance of obtaining items from enemies.
Curators          Critical Master       Increases the chance of landing a critical hit.
Prospectors       Defense Support       Decreases the amount of damage you take.
Outfitters        R&D Support           Increases the number of R&D points you obtain.
Mediators         TP Support            Allows you to accumulate TP even when attacking.

There's a section on Division Rewards as well:

It's good to be part of the club! Once per day, you're eligible for a special reward from your division in the form of consumable items. The precise selection depends on how doggedly you pursued your division's recommended activites. If you want to see how your gang is doing, check out the divisogram in the BLADE Scout Console. Here is a full list of possible division rewards.

Reward            Effect
--------------------------------------------------
Personal Patch    Restores HP to maximum.  Affects all party members but not Skells.
Frame Patch       Restores Skell Frame HP to maximum.  Affects all party skells in operation.
Tensifier         Raises TP by 1,000.  Affects all party members.
Mega Tensifier    Raises TP by 3,000.  Affects all party members.
Fuel Re-Up        Restores 3,000 fuel.  Affects all party Skells in operation.
Special Bonus     Can be sold for a whopping 10,000 credits via the Inventory.
Supreme Bonus     Can be sold for a whopping 30,000 credits via the Inventory.
Salvage Ticket    Salvages a single Skell after its insurance has expired.  
                  Use it at the Skell Garage.