Vampire A with a Stake eventually (through whatever set up you like) inflicts
two stake
damage to vampire B, who plays Undead Persistence.

B receives the Stake (card text on Stake), and is free to use it during the
subsequent rounds
of the persistent combat.

(Same answer for the Rowan Ring)

Note that in the hands of B, the stake's "how much damage have I inflicted on
the opposing
minion" counter starts at 0, since the stake hasn't done any stake damage to A
yet.

At the end of combat, B will go to torpor and will no untap as normal for as
long as he
remains there (even if he gave the stake back to A during the persistent
combat).