S/MIME encryption uses the recipients public key to encrypt an email. Upon receiving the encrypted message the recipient can decrypt it using the private key associated with the public key that was used for encryption.
Why then does the sender need to have public and private key for his/her email in order to send an encrypted message? At least macOS Monterey does not show the options for singing and encrypting with S/MIME unless I select my email address for sending for which I have a certificate in my keychain.