Filecoin, for example, ran their highly successful ICO on the Ethereum blockchain, though they are gonna operate on a different chain.
AFAIK, main-net tokens can't be used on a private chain. Makes me wonder how token sales work for a private blockchain project. I understand that a crowdsale is just another fund raising mechanism, and doesn't even need to be related to a 'blockchain project'. However, if a private chain does use tokens, how would one issue those to crowdsale participants?