The Spoils of War passive is triggered no matter what method is used to kill the minion, but the execute can only be applied with melee basic attacks.
On Summoner's Rift, assuming Eye of the Equinox's user kills the most valuable minions available, it provides:
13.125 per 10 seconds to the user, while also sharing that same gold with the lane partner.
Between 20 and 35 minutes, this is increased to 15.83 per 10.
After 35 minutes, this is futher increased to 22.5 per 10.
Assuming Eye of the Equinox's user kills one minion every 20 seconds, it provides at least 12.5 health regen per 5 seconds to the user and to the lane partner, not taking into account the missing health ratio.
This item was an older iteration of Eye of the Aspect. Unlike the Summoner's Rift variant, it has a unique quest reward, but was replaced with the warding component from Sightstone to allow champions to obtain more diverse build paths.
Reward:Shield Battery, a permanent shield that regenerates slowly outside of combat.
Shield Battery: Grants a permanent shield for 255 − 340 (based on level). The shield regenerates slowly when out-of-combat. Executing minions with Spoils of War regenerates 64 − 85 (based on level) shield value.
Grants +500 health, +100% base health regeneration and +2 gold per 10 seconds.
Unique Passive – Spoils of War: Melee basic attacks execute minions below 400 health. Killing a minion by any means heals yourself and the nearest allied champion for 50 health and grants them kill gold. These effects require a nearby allied champion and consume a charge. Recharges every 30 seconds for a maximum of 4 charges.
Unique Active – Warding: Consumes a charge to place a Stealth Ward that reveals the surrounding area for 150 seconds. Holds up to 4 charges that refill upon visiting the shop.