Should you be concerned about the health of certain trees in your backyard if woodpeckers seem very interested in them, or does it not mean anything?
The trees look OK from the ground but I would not want one of them coming down on the house.
It could indicate an infestation with wood boring insects. Or maybe they just want to put a nest hole there.
if they are moving around on the tree they could be looking for food. if they find food they will stay on the tree. look for holes and bark damage.
What part of the country and what kind of tree? Might be bark beetle epidemic. Or might be they are just making holes to store nuts.
We have a friend who is an arborist. He came out recently and inspected the trees close to our home, did some pruning, and made some recommendations. It was cool; he wore a harness and climbed around like a monkey. Maybe call in the pros if they haven’t been looked at recently and you are concerned.