From Palpatine, the offer is a con. He allows no other to rival him. Vader, on the other hand, seemed to be entirely sincere when he made the offer to Padme and to Luke; but Anakin doesn't want to be in charge himself, he just wants to be important to it.

I disagree with the idea that Sidious is 'stored' inside Palpatine. There is no division between Palpatine and Sidious, they are one and the same person. Palpatine is just Sidious being urbane and cultured to present his actions in the best light when he doesn't quite have the political power to take the velvet glove off his iron fist.