When you use a helpdesk to manage all your customer conversations, all your support queries from the email addresses you integrated will be forwarded to it. However, most helpdesks use a third party app to store incoming emails and send replies.
To be on the safer side, some companies like to use their own servers to send and receive email, by setting up a custom mailbox. These servers could be public services (like Gmail, AOL etc.) or private SMTP and IMAP servers. Most helpdesks also let their customers choose if they want to use their own servers for incoming and outgoing emails or for outgoing emails alone.