I am not condoning shooting a person over a yard sign. It appears (from the news story) that they confronted the person in their yard (driveway) during the theft. The situation escalated somehow because the women's husband was charged with assault. IMO this is not a situation where they just looked out the window saw the person with the yard sign and shot them from the house.
I was making clear that the OP's thread title is a totally incorrect description of what happened. This not about "picking up trash" -- it is about theft -- and deadly force used on the homeowner's property while the perpetrator was caught in the act of the theft. Does this justify deadly force in the situation -- over a yard sign -- well you better check your state laws. Does it justify deadly force if the thief and her companion(s) assaulted the couple in their driveway when confronted? I will let the court system sort the entire thing out. Not enough information at this point.
Stealing a political sign of this type does not reach the level of petty theft. Perhaps trespassing. I had someone try to sell me meat out of their trunk -> told him to leave -> wouldn't move his freezer truck -> wife recorded asking him to leave -> meat guy punched in face -> sheriff called by meat guy -> meat guy charged with misdemeanor trespass.