A power of attorney is never responsible for any debts that the person owes upon death or while they're living.
The power of attorney is only responsible for getting those bills and debts paid from the persons funds or assets and is responsible for making decisions that the person cannot make on their own.
As a power of attorney you do not need to worry about having to pay the debts out of your own pocket when the person dies as that is not the power of attorney's responsibility.