Most legendary effects are pretty straightforward, like "Powerful: +25% damage" or "Assassin's: +50% damage vs humans", Seems like Powerful will always be about half as powerful as Assassin's, but is better for general usage.

Trying to balance these mods, I made a bit of research. We all know (or, having it on read Nukapedia, I do at least) that Two-shot doesn't take Receivers/Capacitators into account when boosting damage, but Powerful do, since Two-shot adds damage by doubling the bullets.

But that's wrong.

In this document I'm going to describe how different Legendary effects affect damage output, with some under-the hood info thrown in.


Some terms:
Base Damage: This is the damage an unmodified, unperked weapon deals. The laser gun has a base energy damage of 24.
Paper Damage: The damage you see on the weapon card in your Pipboy, it is a product of Base damage, weapon mods and perks.
DR, Damage Reduction: this is what reduces the damage done to a target, according to a set formula which can be found on Nukapedia.
DOT, Damage Over Time: A hit from a DOT effect deals damage every second until the duration runs out. Fallout 4 lists the total damage dealt over time and the duration of the effect. DOT effects deal the total damage listed divided by the duration every second (15 D over 3 sec = 3 5D hits, each calculated vs DR separately). May or may not stack.
Stacking: if a DOT effect deals 1D over 10 seconds and stacks, a new hit will not overwrite a previous hit. Hitting twice will deal 2D over 10 sec.
  If a DOT doesn't stack, a new hit will overwrite a previous duration. If the above DOT were non-stacking, one hit would deal 1D over 10 sec, two simultaneous hits would also deal 1D over 10 sec. hitting again after 5 seconds has passed, the total DOT would be 1D over 15 sec.
RoC, Receiver or Capacitator: the primary means of increasing damage through modding a weapon. Some equations use the boost provided, some don't.
Damage types:
For a hit with more than one damage type each damage type is calculated separately against DR, then added up.
BD: Ballistic Damage
ED: Energy Damage
XD: Explosive Damage
PD: Poison Damage
WD: Bleed (Wounding) Damage


Legendary effects and what they do:
Class Damage effects (Assassin's etc): Each hit this effect checks if the target matches a certain parameter (IsAnimal for Hunter's). If the target matches this parameter, the Paper Damage dealt is doubled.
  Increases Paper Damage before DR, so it is affected by perks and receivers/capacitators.

Explosive: This effect only appears on pure Ballistic guns, so any laser or plasma weapon you might have seen with this is created using console or the Legendary Modification mod.
  The Explosive effect changes the projectile of the weapon to apply an explosion on the impact point.
  This results in the Paper damage of the weapon (minus the explosive damage) being applied normally, while each projectile deals damage individually through the triggered explosion.
  Despite the weapon card adding explosive damage directly to the regular damage of the weapon, they are calculated separately.
  This means that if you fire an Explosive Pipe gun that deals 20 ballistic damage and 20 explosive damage (40 damage in the Pip-boy) at a target with 20 DR, each hit will deal 20D vs 20DR + 20D vs 20DR = 20 damage, not 40D vs 20DR ~ 26 damage.
  As the explosion is triggered by each projectile, this can be (and is) exploited on shotguns, which fire eight pellets, resulting in the paper damage from the shotgun plus eight separate explosions, each of which is calculated against the target's DR separately. An Explosive Combat shotgun dealing 50 damage and 15 Explosive damage would hit like this: 50BD vs DR + (15XD vs DR) *8.

  As everyone knows (again, if 'everyone' read Nukapedia) the Heavy Gunner perk increases the damage of the Explosive legendary, probably because it includes weapons like the Missile launcher and the Fat man, which deal very little ballistic damage but a lot of explosive damage. However, in my in-game testing I found that this also applies to Rifleman and Gunslinger. This means that an Explosive pistol normally dealing 15 explosive damage will with 5 ranks in Gunslinger will deal 30 explosive damage, and 45 explosive damage with 5 ranks in both Gunslinger and Explosives expert.
  However, 5 ranks in both Heavy gunner and Gunslinger will not yield 45 explosive damage, as Heavy Gunner does not apply to pistols and vice versa. The same is true for Rifleman.

Freezing: adds 10 energy damage and freezes target on a citical hit.
  If applied to a ballistic gun, the energy damage is calculated vs DR as a separate hit, meaning the extra energy damage will be miniscule against pretty much everything.
  If applied to an energy weapon, the energy damage will be added to the base damage of the weapon. Capacitators will not increase it, but perks will. This means that a Freezing laser pistol will get an extra 20 energy damage with 5 ranks in Gunslinger.
  Shotguns of any type will treat Freezing as part of the weapons damage, this means the energy damage is NOT applied per projectile, it is spread out among the projectiles just like the regular damage of the weapon in the following way: (BD vs DR + ED vs DR)/number of pellets that actually hit.
  Fozen targets take 25% additonal damage from all sources and lasts 10 seconds. This is in the game code but I don't know how accurate this is.

Incendiary: Adds 15 energy damage over 3 seconds.
  As a DOT, this inflicts three 5ED hits, each reduced by DR.
  The effect is added through an enchantment, so the damage will not be boosted by perks.
  The effect does not stack, so even while hosing an enemy this effect will still only deal 5ED vs DR per second.

Plasma Infused: Works exactly like Freezing, except instead of freezing on a critical hit any target killed by a weapon with this effect may have Nuclear material in its inventory.

Instigating: doubles damage dealt against targets with full hp.
  This effect works exactly like Class Damage, except it checks whether or not hp is 100%.

Junkie: Weapon deals +15% Damage per addiction.
  This effect works exactly like Class Damage, except it checks how many addtcions the player has and adds a damage bonus percentage equal to 15 * AddictionCount.
  Minimum is 0% bonus at 0 addictions, maximum is 165% bonus at 11 addictions (with massive SPECIAL penalties).
  Depending on what stats you value, you should be able to deal with 5-7 addictions without crippling your character.

Lucky Weapon: Critical shots do double damage and critical meter fills 15% faster.
  First off: The description is a lie.
  Second off: I don't know if Nukapedia is accurate when it comes to crit damage, or if RoC or weapon perks part of the equation. I will assume Nukapedia is correct in this instance.
  Critical Damage is additive. A regular crit will add +100% Base Damage to Paper Damage, Lucky adds another +100% Base Damage.
  The Better Criticals perk adds a maximum of +150% Base Damage (+50% per rank), meaning a critical hit with a Lucky weapon and 3 ranks in Better criticals will deal the guns Paper Damage + Base Damage * 3.5, or 350% increased damage.
  A Calibrated receiver will tack another +100% onto that, alternatively a Photon agitator adds +200%, for a max total critical hit that deals Paper Damage + 550% of Base Damage to a target.

Each hit does not fill the crit meter (CM) 15% faster, it fills the CM by 15%. Take into account that at 1 Luck, each hit fills the CM by 7%. This means that at 1 Luck, a Lucky weapon will fill the CM by 15+7=22% each hit, filling the CM in five hits. This is equal to a +10 Luck bonus.

An image with most of this information exists in the image section of LMR.

Penetrating: Ignores 30% of targets DR.
  The way DR works, this essentially increases Damage dealt against targets by a flat 14%, unless the DR is very low in which case it does nothing.

Poisoner's: 30PD over 10 sec.
  Works exactly like Incendiary, but Poison Damage requires its own DR, Poison resistance. Everything in Fallout 4 has Poison Immunity except humans, who only has poison resist if they're wearing a Poisoner's Legendary armor piece.
  Poisoner's DOT doesn't stack, so you must wait ten seconds between each hit if you want the effect to deal its full damage.

Powerful: +25% Damage.
  This bonus applies to Base Damage * Perk bonus, ignoring RoC.
  A Powerful weapon dealing 50D with 5 ranks in its weapon perk will deal 100D+25D=125D. Add a Powerful/Maximized RoC (+50%D) and the result will be 100D*1.5+25D=175D, not (100D+25D)*1.5=188D.

Two-shot: Fires an additional projectile (also double damage, double recoil and double spread).
  The number of projectiles and the bonus damage are separate entries and can be changed individually using an editor (like the CK or FO4Edit).
  Two-shot adds one extra projectile, meaning ballistic shotguns fire nine pellets instead of eight.
  The damage bonus works exactly like Powerful.

Wounding: 25WD over 5 sec.
  Deals bleed damage, which has no DR associated with it. It will therefore deal its full unmigitated damage against any target.
  Stacks, so if you hit a target ten times in one second, the effect will
deal 250WD during the following 4-5 seconds irrespective of target DR.
  Applied per projectile, meaning that shotguns can apply one bleed effect per pellet.
  Can be found on any weapon type.

I can look into the mechanics of other Legendary effects if requested.