Additional Brains are completely separate Organ trees. All brains are active at the same time.
If you were wondering how to actually use bladders, this is how. Bladders are great Organs to start out your 2nd or 3rd brain set.

Activates the right-side Organ on Death.
Death of the left-side Organ activates the right-side Organ. Meaning that Offspring will trigger this organ on death. Incompatible with Brain unlike the other Hearts.

Disclaimer: This is unverified second hand information from multiple comments.
Activate the right-side Organ when the left-side Offspring or the Brain takes damage
If attached to an Offspring, specifically the Imp, will activate the right-side Organ each time the Offspring takes damage. This is possibly the highest efficiency and activation rate in the game depending on how fast you can damage the Imps.
If attached to the brain, will activate the right-side Organ whenever the player takes damage.

Sometimes when chaining organs together you may run into an Insufficient Duration warning.
This happens when the activation rate of the current organ is too slow for the duration of the left side organ.
Effectively, if you have a projectile that lasts 1 second but your following organ only triggers at a rate of every 1.01 seconds, the organ will not be active, nor any organs further down the chain.
This Is because every link in the chain is dependent on the previous link to activate. If a projectile does not last long enough for the next link to activate then the link won't be able to activate.

The best advice I can give you when making long chains is to use items with Pierce: ∞ and long projectile duration early in the chain.

All Intestines create an aura of the right-side Organ on the ground that does not move.
If connected to the Brain, they will spawn the aura at the Player's position when triggered.
If connected to an Organ, they will spawn the aura at the Organ's position when triggered.
The aura lasts the same duration as the right-side organ and rapidly stacks the right-side organ's effects to targets in the radius.

All lungs create an aura of the right-side organ that is centered on and follows the position of the left-side organ.
If connected to the Brain, the aura will follow the Player.
If connected to an Organ, the aura will follow the Organ.
The aura lasts the same duration as the right-side organ and rapidly stacks the right-side organ's effects to targets in the radius.

Replacing Offspring before their duration runs out counts as killing them.

Each Offspring spawned is it's own individual and activates it's right-side Organs at a rate unique to itself. Because of this Offspring will effectively stagger their right-side Organs' activation by the rate at which they are spawned, leading to a much more consistent overall rate of activation between them.

The most important thing for Wings affecting the player is efficiency. Bladders are the best option for this. Bladder of the Kangaroo and Wings of the Fairy are an excellent example of a simple and synergistic pair, but there are plenty of others for you to find.

One way to use Wings is with Lungs or Intestines. This will apply Wings to all friendly units in the AOE including yourself. This is very good for swarm builds because anything you chain after the wings will proc on each unit affected by the Wings. For example this means that you could chain an Intestine into a Wing into a Jackal Spine into a Starfish and each unit affected by the Wings will become a Starfish turret independent of the unit's other operations.

Keep in mind that most AOE Wings will not have the best efficiency and as such are really only useful when chaining other effects besides their own.

As of the writing of this guide I have yet to fully verify what all the interactions of the following Organs are:
Offspring of the Imp
Bladder of the Frog (specifically what effects are considered a Debuff)
Heart of the Wolf (specifically how it interacts with Imps)