Because everyone knows that private key is used to spend, and public is used for verifying balance, but not many people really konws how does it really run, I'll try explaining it.
Private key, for signing transatcions
Private key is never transmitted to the bitcoin network. There is only used for signing transactions and for generating public key.
Public key for verifying signature
Public key is derivated from private, and is used for verifying signature. For example, if alice sends 1 BTC to bob, she will create a transation and sign it with her private key. Bob will only need for Alice's public key, transaction and transaction signature to verify.
tip : We can use message instead of transaction. Transactions are just messages with money transfer instructions.
