It depends on your phone or OS, or message app. I have persistent notifications without doing anything to get them. I also use the TeslaUnread utility with Nova launcher so get unread counts in badges on the icons.
It may also depend on what you mean by notifications. LED notifications don't persist indefinitely, but my notification LED is so small that it's barely useful anyway. And it doesn't keep making a sound (though I could set my message app, Textra, to repeat notifications if I wanted). But notifications in the notification bar/pull down/lock screen persist.