For messages sent through an insecure channel, a properly implemented digital signature gives the receiver reason to believe the message was sent by the claimed sender. While using a digital signature

The sender's public key is used by recipients to verify (decrypt) the signature, not to create it - encrypting with the public key would allow anyone to forge the signature.

The receiver's private key is used to decrypt confidential messages sent to them in asymmetric encryption, and plays no role in validating or creating a digital signature.

The receiver's public key is used to encrypt data intended for the receiver's eyes only, which is a confidentiality operation entirely separate from digital signature authentication.

The sender uses their own private key to encrypt the message digest, creating the digital signature. The recipient then decrypts the digest using the sender's corresponding public key - if decryption succeeds and the digest matches, it proves the message was signed by the entity holding that private key and has not been tampered with in transit.