I have moved my bitcoins around many, many times. I have never been asked to produce a private key, and I don't even know where to find them. Is that wrong?
For most users, that's perfectly fine. The software you use manages your private keys for you and stores them in the wallet-file. If you use any of the standard Bitcoin wallet clients (Bitcoin Core, Multibit, Electrum, etc...) that's good.
If you use a web-service to manage your Bitcoins, then you don't have your private keys on your computer and you're 100% reliant on the web-service to keep your money safe. This is not recommended (except for small amounts for convenience).