That, unfortunately, depends on the app/service you're using. Some apps don't sync read receipts across devices.
For instance, GMail does, so you will get the notification on all devices, but if you read it or delete it in one, within a couple minutes (depending on your connections), the notification goes away in all devices.
Other apps, like Facebook's Messenger, will not clear the notification if you've read it elsewhere (this seems to be buggy, though, sometimes it does, most times it doesn't... and what's more, even if you didn't have your device ON and you got messages and already read them, most of the time the other devices will still get those flooding in once the device is powered back on).