Since both use the same mail address?
If both PC's use the same email address/account, you should see all your emails on both exactly the same.
For whatever reason you want to send an email to yourself, you can do on either PC and you will see it on both.
Example:
If abc123 (at) email dot com is your email address, if you send an email from that address to the same address, both PC's will show the same.
(or, have I misunderstood your question?)