For one person, sometimes Sent Items don't show up and he doesn't have a record that he sent that message unless he 1) cc:'s himself, or 2) someone replies with his email in the thread. This isn't happening all the time and when it has happened, he'll simply wait and then the Sent Items will start populating.
I spoke to him today when it happened. I sent a message to myself from his Outlook with subject line "Testing at 11:15 AM". I got the message but the Sent Items never showed the message. The view was set to show newest items first and there was no filter. I even went to View -> Reset View but there was no change.
The test email doesn't show up in Sent Items in OWA or mobile either. I checked to make sure that the status bar showed "All folders are up to date. Connected to: Microsoft Exchange." I checked rules and doesn't have a single one. Because this is happening in OWA and mobile as well, the issue sure seems to be on the Exchange Online end. We're using Microsoft 365 services.
He has Microsoft 365 Apps for Enterprise version 16.0.14026.20270 last updated 6/14/2021. No one else in the small organization is reporting this and even for him, the issue will occur without any warnings or errors and the Sent Items will just simply start working again at some point and he'll have a record of messages that he's sent.
He notes that items that disappeared never come back. If he sends messages and they don't appear in Sent Items they never appear even later when things appear normal again. I spoke to him a few hours later and items are appearing in Sent Items again but that "Testing at 11:15" message never appeared.