How do notifications work internally?

Hi, @Mikel WHIPPLE 

Q: I have a notification set to notify assignee and group field for the "you're assigned a work item", if there is no assignee will it notify everyone in the group?
A: Yes, it will notify the group. Notification will be sent to everyone who you added in the Notification scheme for that scenario, as long as they have Browse Projects permission on the issue.

Q: If there is an assignee and a group, will it notify both?
A: Yes, both will be notified. Notification will be sent to everyone who you added in the Notification scheme for that scenario. If a user matches both entries (assignee and group member), Jira deduplicates and sends only one email.

Q: What if I am in the group and change it to my group, will it notify me and the group or just the rest of the group, or none of us?
A: The rest of the group will be notified, but you will be skipped by default. Jira Cloud doesn't notify you about your own changes unless you enable the option "You make changes" in Personal settings > Email notifications for issue activity.

Q: Also, where can I see all the outgoing emails at? I am seeing some team members notified, and some not and they are very much in that group.
A: I'm afraid it's not possible. Cloud instance doesn't offer an outgoing email log, only Atlassian Support can check actual delivery on their side. But you can use Notification Helper (Settings > System > Notification helper) to check whether a specific user would receive a notification for a specific issue and event, and why not. Most common reasons for "in the group but not notified" are missing Browse permission, personal notification preferences, or the user being the one who made the change.

Also, it's worth to look at 
https://your-company.atlassian.net/jira/settings/personal/notifications
There user can configure, what he want/don't want to receive. It overrides system settings.

Hi @Mikel WHIPPLE 

Each recipient type you add to a notification scheme event (e.g. "Assignee" and "Group Custom Field Value") is evaluated independently, so here's how your scenarios play out:

No assignee, only the group configured: Only the group gets notified. Jira doesn't fall back to notifying the whole group in place of a missing assignee  an empty "Assignee" recipient entry simply sends nothing, while the "Group Custom Field Value" entry sends to everyone in whatever group is set in that field, regardless of what's in Assignee.

Both an assignee and a group configured: Both fire. The assignee gets notified via the Assignee recipient rule, and everyone in the group gets notified via the Group Custom Field Value rule, independently of each other.

You're in the group and you're the one who sets it to your group: By default, Jira does not notify a user of their own changes  "My changes" is set to "do not notify me" out of the box. So you personally won't get an email for your own edit, but every other member of that group will, exactly as normal. This is a per-user setting though (Profile → Notifications), so if a teammate has switched their own-changes setting to "notify me," they'd see their own actions too, that's likely why "some team members are notified, some aren't" even though they're in the same group: check each affected user's personal notification preferences first, not the scheme itself.

To see why a specific user did/didn't get notified for a specific issue/event: Use the Notification Helper,  ⚙ Settings → System → Notification Helper. Give it the issue, the event, and the username, and it tells you whether that user is currently eligible to receive that notification and why (or why not). It's the standard first stop for exactly this kind of "some people got it, some didn't" mystery. Note it only confirms eligibility, not that the email was actually delivered (mail server/spam issues are separate).

Reference:

• https://support.atlassian.com/jira-cloud-administration/docs/configure-notification-schemes/
• https://confluence.atlassian.com/adminjiracloud/jira-admin-helper-818578850.html