When you tell Enigmail to OpenPGP-encrypt an outgoing message, Enigmail should automatically select the recipient's public key from your public key ring. If you don't have a matching public key in your keyring (by default, Enigmail selects keys via matching email address), or if your "web of trust" doesn't sufficiently trust the recipient's key, then Enigmail will pop up a dialog box allowing you to explicitly choose the public key to use (or to cancel sending, to give you time to obtain a suitable public key, or to sort out the trust of a key you already have).
It is highly unusual to use OpenPGP to encrypt outgoing messages with your own private key. In fact, if that's the way you usually use asymmetric encryption, then you're doing it wrong. The common idiom is that public keys are used for encryption and signature verification, while private keys are for decryption and signing.