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Allow notifications: untoggle this, and you will never see alerts from this applications, meaning that you will have to actually look inside the application to see if you have received a message.

Alerts: you can leave notifications on but deselect the Lock Screen option, so you will only see the message alert in your iPhone Notification Center.

Show Previews: if you leave lock screen notifications on, you can make sure that only you can view them after unlocking your device with Face ID, or that they can never be opened.

Sounds: Use this menu to turn off the default alert tone and vibration for the application, so your phone would not give it away when you receive a text message from it.

You may only want to hide one specific conversation's alert notifications. How to do this would vary from application to application, but it is relatively very foward in Apple Messages. Open up the conversation, tap on the profile icon on top, tap Info, and toggle the Hide Alerts option.

How To Hide A Conversation From Apple Messages

Hiding alerts from a very specific contact works to an extent, but the text conversation would still be fully visible in-app if someone else unlocks your device, especially if you texted that particular person recently.

The most straightforward way to hide a conversation is to delete it. Either tap and hold on a Messages convo or swipe from right-to-left to make the trash icon come up. Then tap the red trash icon to confirm that you want to erase it.

If you want to save your messages before you delete them, back up your iPhone to your computer and then use the Decipher Textmessage application for Mac or Windows. It reads your iPhone/ iPad backup and exports your conversation as a PDF file, multimedia version with photos and video, plain text version, or other options.

Assuming you do not want to delete the conversation and want to continue texting this person, your best bet is actually to delete their contact from your address book. Then, open up Settings > Messages and scroll down to Message Filtering. From there, you can toggle Filter Unknown Senders.

Finding A More Private iOS Messaging App

Apple Messages was designed for ease of accessibility and convenience, not necessarily for privacy. Instead of using workarounds and disabling features, you should consider switching to a new application that prioritizes privacy instead.

Messaging applications like Signal and Telegram offer end-to-end encryption so your messages cannot be hacked. More importantly for this discussion, they allow you to password protect the application itself, so someone with access to your phone password still cannot access your private messages.

Another neat privacy feature is that some messaging apps let you to activate Auto Deletion for specific conversations. Telegram, for example, lets you set a timer of 24 hours or 7 days, then you can have any messages older than that get deleted by default.