You need to teach him “leave it.”
Dogs work on the function that everything is theirs unless the Alpha claims it. You and your SO need to be the alpha and start claiming things — A dog should not own anything unless it is given to them, and even then it is only on loan from the Alpha.
Basically, start off small with a treat. Show the dog the treat and put it on the floor. PUT YOUR HAND OR FOOT ON IT. This is very important. You are taking ownership of this treat. Remove your body part and tell the dog to leave it. If the dog goes for the treat, immediately remove it from their mouth, sternly scold them (I use the word “Wrong” but use whatever you normally use) and put the treat back and repeat the process.
When the dog successfully leaves it for a few seconds (start off slow), take the treat and hold it to your chest, praise the dog and say “Share” and give them the treat.
This will start to make the connection. When you say “Leave it”, that means “Leave it alone.” Leave it will work for People, Places and Things. This is good when walking. Your dog wants to investigate a mailbox? Leave it. Wants to chase a cat? Leave it. Wants to be over friendly with a house guest? Leave it.
It also means that when you say “Share”, you are surrendering ownership of the item to the dog.
Start off with treats and work your way up. Never put your dog in a position to fail (Such as waiting for him to go for the trash and then scolding him), so you can approach the trash with him, call his attention to it, and then tell him to leave it.
I usually couple my commands with hand gestures. When my dog is told to sit, I flick a finger upwards. I can command him silently if he sees the gesture. If I tell him down (“lie down”) I point to the ground. Same when I tell him off (“get off the couch/person/etc”) or when I send him to bed or out of the room. For “Leave it” I usually wave my hand over the object or between him & the object.
I have a nearly 2 year old hyperactive black lab and he has taken to “leave it” exceptionally well. Labs are smart and attentive. It took my dog two weeks to be great at leaving things. If I put a treat on the floor, instruct him to leave it, and leave the room? When I come back, it will still be there.