Who keeps a copy of the POA, and for how long?

There is NO POA on death.

Once this person passes, their financial (etc) affairs are controlled by the trust if there is one, or the will under the direction of the executor. POA ends with death

I have the only original copy of my mother's POA. Several banks have their versions of is on file. Even though the POA ends upon my mother' death, I will probably hold onto it for a while in case there are any questions but technically, I guess you could get rid of it when it is rescinded, replaced by a new POA or upon death of the principal