These are custom notifications that are setup in CRM, found under Administration > Notifications. You set it to run with an ad-hoc query, and any constituent that is returned by that query gets that notification on it.

The sync related notifications are related to what kind of sync errors they have. Look up the ad-hoc query to see exactly what kind of errors it is looking for. If an account in question does not have that kind of error, then it will not have a notification in the constituent.

In one case, in TA, a synced account had been inactivated.  In CRM, this record showed that there is a Sync Conflict.  This means that synced records have been updated in both systems since it was last synced. 

In order to correct these errors, go to the "Team Approach Sync Conflicts" screen in CRM and find each Account ID by the "TA ID 1" column:  
Administration -> Team Approach sync (under Administration) -> Team Approach sync conflicts link -> Unresolved Conflicts section

After locating the Account ID row, click on the double down arrows and then select whether you want to overwrite TA or BBEC for a particular row.  

In this case, since the records are supposed to be Inactive, selecting the "Overwriting BBEC" makes sense.