If I have POA, and my name is on the checking accounts as POA, what happens when the person dies?

My name was on my Dad's checking account so that I could also sign checks for him. The bank said that Dad had me as beneficiary, thus everything in that account now belongs to me, and doesn't need to be put through probate.

Therefore, ask the bank how the checking account was set up, and who is the beneficiary named.

For myself, I plan to wait until probate is over to touch any money in that checking account in case there is a question. 

Unless you are named on the account as the beneficiary or joint owner (not just POA)

POA ends at death. You have no further access. The Executor of the Will takes over.