It all boils down to maturity and trust of your play group as well as the implied (or even stated) social contract.
What I mean by that is it is circumstantial. Can you trust your player to play as if they were charmed? Will they act indignant or get angry about it? As Chad stated, can you count on the player to properly RP the situation like they RP their character on a normal basis?
You have two options at this point once you have answered the above questions:
Trust your players: This is assuming that your players are mature enough (or even have the skill) to RP their characters while charmed, regardless of where the charm comes from. This is the best of the options in that your charmed player doesn't feel like they are out of the game and watching their friends/party members have fun without them.
Play the charmed PC yourself: This is the worst case scenario of the two options. You are effectively making the PC an NPC (or GMPC) for the time being and isolating your charmed player from play. This is no fun for them and has the potential to be a negative gaming experience for your group. This does imply a lack of trust with your players and in many ways violates the implied social contract of your group.
All of that said, if you choose to allow your player to play the charmed PC, bring them over to your side of the table, couch, whatever and have them work with you. If that means you are whispering in their ear for their response to something, so be it. They feel involved and might even have fun potentially conspiring against their friends and fellow players. If this is a PbP or PbE game, send them a private message with some potential responses and let them decide how to respond.