It seems there is still some confusion on what Insight exactly does. It has many uses, though it's utility has been tremendously overestimated at launch. There was much speculation, but only the following things are actually affected by Insight (note I am talking about the Insight stat on your hud that ranges from 0 to 99).
- Witch of Hemwick will not spawn Mad Ones during her fight.
- The Plain Doll comes to life, allowing you to level up your character.
- Beckoning Bell and Silencing Blank are received through the Messengers in the Hunter's Dream.
- Insight Bath Messengers shop becomes available.
- Lamp Pimps in Cathedral Ward gain a projectile attack.
- Scythe Pimp in Cathedral Ward gains a weapon buff.
- Cross Pimp in front of the Grand Cathedral gains a small passive AOE frenzy, in addition to a weapon buff.
- Mad Ones (the enemies Witch of Hemwick summons) will spawn in Hemwick Carnal Lane if you have learned the password.
- Gehrman mumbles in his sleep in Hunter's Dream.
- Brain enemies start singing off-key.
- Lesser Amygdala's are visible before you defeat Rom.
- Hunter's Dream music changes before you defeat Rom.
- A crying baby can be heard occasionally.
- The higher your Insight, the lower your frenzy resistance will be (resulting in becoming frenzied faster).
- The higher your Insight, the lower your Beasthood will be (resulting in dealing less damage after using Beasthood Pellets or innate Beasthood weapons in relation to if your Insight was lower).
Note: Most visual and ambiance changes occur automatically after defeating Rom and Insight will have no affect on this anymore.
That's all there is. There is some speculation whether or not the enemies in Lecture Hall may gain additional attacks at higher Insight, but it's unproven. Otherwise, much of the other rumors has to do with the fallacy of people assuming Insight is responsible for what happened without actually testing it thoroughly. For instance, the Amalia dialogue that was supposed to happen at high Insight also happens at an Insight of only 2. The difference was the person with high Insight walked slowly towards her and assumed his high Insight was the cause. Instead, you can slowly walk up to her (before she transforms) and the dialogue triggers no matter what Insight you have.
That's not to say this is a definitive list, some stuff might still be undiscovered. But things like damage increases, damage decreases, boss moves/damage and whatnot have been thoroughly tested and debunked.