I do see emails i forward or reply to in my Sent folder . thats is the sent folder in Email Inbox , NOT in any other mail client. I asked this question during the V8 email inbox webinar which they confirmed
If in your ERPNext account, the Email Inbox is configured for one account, but as per the Email Account, default outgoing email gateway is set to another,. This could be the reason, sent emails are not appearing in the Email Inbox view. In my test account, since my ID, email inbox and outgoing email gateway is all the same, it is correctly showing sent emails in the inbox view as well.
Another reason could be for the copy of emails on the server, you should look for the setting of your email service provider. In Gmail, it is provided by default. In other services like WebMail, you have to specifically enable the feature of leaving a copy of an email on the email server always. Perhaps you can check the setting of your emails service, as it must be configured at that end.