Where does it go and who verifies private keys?

The private key is used when signing a transaction. The private key usually goes into a bitcoin wallet, and the wallet software handles the signing and storage of your keys. Do not give your keys to anyone. Anyone with your private keys can spend just the same as you so it is important to keep these safe and preferably offline.

Yes your wallet software will verify the signature (not the key) before broadcasting as well as any listening nodes.