I'm new to bitcoin and I still can't find a clear answer to my doubt.

If I buy bitcoins on an exchange which won't disclose my private keys, it means I can't send the coins to an address of, say, my electrum cold wallet?
Is this the case? I would be forced to sell and buy only within the exchange? 
If so, how can I use my coins for something else than trading or investing? 

The next question is, is there a list of exchanges that disclose your own private keys?