THE PENGUIN ALWAYS TAKES HIS CUT: The Penguin is a resource assigned to your force and is not placed on the map. Once, at the beginning of your turn, you may choose a friendly character. That character can use the power indicated on the dial's current click and modifies its combat values as per the current click. At the end of your turn, if the chosen character was given a non-free action this turn, roll a d6 and compare:
1 = Remove The Penguin from the game and heal the chosen character 1 click.
2-5 = Turn the Penguin's dial to the right once. If The Penguin is KO'd it is removed from the game.
6 = Turn The Penguin to any click on the resource dial.

Assuming the penguin isn't KO'd, how long does the recipient continue to modify stats and have access to the showing power(s)? You choose the character at the beginning of a turn, and at the end of the turn you roll a die, but it doesn't specify when the selected character stops using the penguin. Is it until you select a new target? If you select a new character, does the previous character stop gaining penguin bonuses? It doesn't say only one character can use it at a time, only one character per turn can be selected.

I apologize if there is a post about this already. I searched but couldn't find anything specifically to this subject. I also couldn't find anything in the player's guide. Hopefully I'm not just blind

Until you choose another character.

It doesn't need to say when it stops; if the character isn't the chosen character you chose at the beginning of that turn, then it is no longer "that character". And only "that character" gets to use the resource.

Interesting, I had assumed the Penguin bonuses only lasted until the beginning of your next turn as I thought that was consistent with previous "choose at the beginning" effects, though on further consideration I can't think of any offhand that don't come with a qualifier like "until your next turn" or "this turn."