It's a good idea to hang onto them for a short time in case you have to refer to them in further correspondence. But stored messages can build up alarmingly fast, and it's necessary to go through every now and then and clean out the old stuff. In summary then, outgoing mail sits on the originator's computer until the writer takes action to send it. It's then passed through as quickly as possible until it ends up on the recipient's host computer, where it rests until the recipient takes action to retrieve it. Mail is then deleted from the recipient's host, and the only copies remaining are in the personal computers of the originator and the recipient.
Nothing is kept on the Internet itself.
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