What does notify anyway mean on Do Not Disturb?
As an iOS user, you may have noticed a notification underneath a sent iMessage stating “This person has notifications silenced” with a hyperlink underneath giving you the option to “Notify Anyway”. This option bypasses a person's notification settings and lets Notify Anyway iOS 15 messages ping through.
It indicates that you received the message but did not get a notification, like a vibration, sound, or banner. Notify Anyway: This will appear under the “notifications silenced” alert. This is, in fact, a button. This lets the sender override your notification settings.
Apple's Focus feature tells other iPhone users when you're using Do Not Disturb mode, giving them the option to notify you anyway.
So, if the Reminders and Messages apps are not in your list of Allowed Apps, Do Not Disturb will still let their notifications pass through due to the nature of their content. If you don't want to receive these, go to your Do Not Disturb settings in the Settings app.
Allow Someone to Call You in Do Not Disturb Mode
On Do Not Disturb screen, scroll down and select Calls, messages and conversations option, located under “Exceptions” section. On the next screen, tap on Calls and select Favorite contacts only option in the menu that appears.
Will You Still Get Texts on Do Not Disturb? Yes, you will continue receiving texts. Do Not Disturb mode on iOS and Android prevents notifications from making noise or appearing on screen. It does not block incoming text messages or phone calls.
If a chat says notifications were silenced, it doesn't mean that that contact blocked you. They might have turned on Focus mode on one of their Apple devices which will silence all notifications throughout all their devices.
What does “Notifications Silenced” in Messages mean? As I stated above, this means that the recipient of your message is using Focus and she or he has notifications silenced. If you are using this feature, then your friends may see this message when they message you.
The call will go to voicemail, but won't ring to distract you. If you are on a cell phone, you can set your DND to allow certain calls through.
“The last message that you know was received and responded to…should have a 'delivered' status,” Lavelle explains. If, under the messages you've sent after that, you don't see that “delivered” notification, that could mean that person has blocked you.
How do you get past someone's Do Not Disturb on iPhone?
A separate option for calls and the only way to pass through text notifications required a non-obvious process: open a contact, tap Edit, tap Ringtone (for calls) or Text Tone (for texts), and then tap Emergency Bypass. You can make this change in either or both Ringtone or Text Tone. Tap Done when complete.
She was referring to putting her phone on “do not disturb” mode to ignore incoming texts from the sender — so named for the little half-moon symbol that signifies “do not disturb” on the iPhone. The iPhone's Do Not Disturb mode silences everyone, temporarily turning off all of your notifications.

Generally, Do No Disturb will not allow calls. However, if you wish to receive calls from specific people, you can go to the DND settings on your phone and choose ( 'star') the contacts you do not wish to block.
You will still get notifications about missed calls (unless if you have disabled them). But to receive calls you must turn off DND on your mobile device.
Texting someone who's blocked you works as you would expect. The message sends as normal, and you don't get an error message. This is no help at all for clues. If you have an iPhone and try to send an iMessage to someone who has blocked you, it will remain blue (which means it's still an iMessage).
Blocked text messages disappear
When someone that you've blocked texts you, their texts go nowhere. They won't receive any notification that they're blocked, and their messages will still look like they've been sent.
What does 'Notify Anyway' do? This is an SOS option of sorts introduced by Apple that lets you bypass an active Focus mode on your contact's device and deliver important messages. This option can be used in emergency cases where you really need to get a hold of someone.
The threat isn't real, and scammers simply seek to gather personal and financial information to defraud you. Apple never uses security alerts on its devices, so whenever you encounter one, you can be sure it's not legitimate.
If you have an iPhone, iPad, or iPod touch, Find My will send a notification to your Apple device. This feature is available on iOS or iPadOS 14.5 or later. To receive alerts, make sure that you: Go to Settings > Privacy > Location Services, and turn Location Services on.