Okay, here's everything I have been able to gather from my own tests and those of others in this and other communities:
The buddy with you gives you a 5% proc chance per level, so anywhere between 10-50% even if they're the only character you have specialized in that skill.
Each inactive friend with the same specialization gives a 1% bonus per level, so anywhere between 2-10% each.
Regardless of how many items you get from an action (like 3 apples per harvest), your Hangout Buddy always drops a single bonus item, with one caveat. If your proc chance gets above 100% (such as with 6 level-10 Friends), you then get the remainder as a chance of getting a second item instead of just a single one, but you get a 100% chance of the bonus item every time you trigger it.
Triggers for Friendship bonus items are as follows:
Digging: Digging a new hole that isn't glowing and getting an item from doing so. The chances of getting something (coins, earth, clay, etc.) seem to be about 50%, so about half the holes you dig with a Digging friend will have a chance to grant you a bonus item. By far the most RNG-dependent of the five skills. Digging the sparkling patches of earth that drop Dreamlight Shards and Night Shards do not appear to proc bonuses. Filling in an existing hole or redigging a hole that you've filled in manually also does nothing.
Fishing: Completing the last real-time cue for a given catch (be it a fish or seaweed). The bonus doesn't trigger upon picking up red burst bonus fish, but whether you're fishing a bubble or open water, any time you reel something in, your buddy can toss you an extra.
Foraging: Any time you take the "Harvest" action to gather fruit or herbs/spices/mushrooms your buddy can double it. This includes however many triggers you can spam into the duration of a golden crop. It does not trigger when you pick up red burst crops. It also doesn't trigger when you use the "Pick Up" action to forage flowers or wood.
Gardening: Your buddy has a chance of giving you a bonus item (or more above 100% chance) per "Harvest" action of seeds you planted (not the ones in WALL•E's garden), including bonus harvests of glowing crops. Picking up red burst crops does not trigger this potential bonus.
Mining: A Mining Friend can toss you an extra unit of the rarest material dropped from a given rock pile on your final swing of the pickaxe (when the pile crumbles and can't be interacted with again until respawn). If you got a gem during mining that node, you will always get a bonus gem; if you only got other minerals, you'll get the one worth the most in resale value. Picking up red burst loot drops does not have a chance to trigger this bonus.
Other actions that do not trigger a Friendship Bonus item are:
- Removing night thorns
- Breaking rock obstacles
- Dissolving mushroom obstacles
- Digging up stump obstacles
- Picking up a memory
- Approaching or feeding a Critter
- Crafting an item at a workbench
- Cooking a meal at a stove or campfire