Two stories:
First one-
One night I'm sitting on the couch and my husband looks at me and his eyes go big as saucers. "Don't move!" he says and runs away. Real comforting. So as the worst scenarios are going through my head, he comes back with gloves and a stick and a big box. Great. I was imagining snake, so possum was better, but not by much since it was behind me. It crawled under and into my couch, and we had to flip it over and prod it out. It was fairly young and stupid (like most of us were in our adolescence), and we re-located it. It had gone into the dark garage when the kids left the doors open, then to the kitchen, then living room. I'm just glad we found it before it ate a pet or made a home in my couch!
Then a few years later-
One night I heard something in the attic, and "encouraged" my husband to go investigate. Nothin' doin' at 3 am. Next night the same. And the next. So I'm the only one who ever hears it, and I was pregnant and having weird dreams, so hubby blows it off. This goes on for months. Then come Spring we see Momma Possum and her eight babies crawling out from the eaves of our roof. My kids were impressed, "She has more kids than you!" Nice. I was just glad I wasn't crazy with the weird noise issue.
We re-located them (hubby pulled them out by their tails) to the back woods, and haven't seen them since. We didn't have chickens at the time, though, so if we had I might have dispatched them instead. One reason we didn't kill them is that we have snakes and they eat them. I found 3 coral snakes in my front flower bed alone! True red-and-yella-kill-a-fella coral snakes! For us, that was a good trade at the time to get rid of venomous coral snakes and copperheads. But like I said, we didn't have chickens at the time either.
I don't think you should go straight to killing if you can avoid it, but I feel the same way about possums as I do about criminals. As long as you abide by the law and don't take someone's life you get to keep yours. When you go on a killing spree, you lose your right to be part of society.
As for disease, I had a vet tell me that they are one of the dirtiest animals in the animal kingdom. They are carrion eaters with all that implies, and carry everything you'd never want to have from viruses to bacteria, worms, and parasites, and they are known to carry diseases like leptospirosis, tuberculosis, relapsing fever, tularemia, spotted fever, toxoplasmosis, coccidiosis, and trichomoniasis.
Google it and you'll never touch one with your bare hand, much less let it near your chicken coop. To the guy who replaced his chicken's food and water, that was smart. Their feces is usually shedding worms, parasites, and the like.
Ok, possum stories over. They were cuter when I didn't have chickens.