No. 1:1 emails sent from a contact's profile or an open deal will NOT appear in the sender's "Sent" folder of their email client.
The only time this can happen is when a user's email is connected via Settings > Deals (Sales Plus and up plans). Once connected, only Automation 1:1 emails will appear in the "Sent" folder of their email client.
Note that users on the Marketing plan can send 1:1 emails to contacts via contact record and contacts overview page but it will not appear in the "Sent" folder.