Is this 1 Microsoft Account with 2 email addresses or 2 separate Microsoft Accounts?
To see the email addresses associated with your Microsoft Account, go to;
https://account.live.com/names/manage
Regarding the Sent Items themselves;
- Do you see any Sent Items at all?
- Do you see Sent Items for the other address?
- When you sent a new message, does the item get stored in the Sent Items folder?
- Are you using Outlook on the Web for this or are you using your Outlook.com mailbox in an App? If yes, which one?
When using Classic Outlook, make sure you have the option selected to save sent messages;
File-> Options-> Mail-> section: Save messages-> Save copies of messages in the Sent Items folder