I'm really surprised they haven't added a way to do this. It's been a super annoying feature ever since buff servers were more prevalent and people would pick up and drop scrolls to attract masses of PvE players to farm. Even recently, I changed my home campaign just last week because people were doing that.
I accomplish this, and some other things, with an addon called No Thank You. I would post the URL but my phone browser refuses to let me paste the URL into this post. You will have to google it. It is available at ESOUI.com.
There are SO MANY other things that need to be added/fixed before they even consider something minor like this. I wouldn't even want them wasting ANY resources to add something like this when there are literally hundreds of other things that should have priority over this.
Please... please... please. I go into PvP rarely - at most, once a week - and I am really tired of the constant pop-ups. All I want is an option to turn the announcements off, as we can already do with Leaderboard shouts.
The updates were useful since it let you know if some of your Cyrodiil bonuses were in jeopardy, but now that they don't apply outside of Cyrodiil it would not be an entirely bad idea to prevent the messages from displaying outside of Cyrodiil as well.
Guildmaster of Power With Numbers in PS4 NA Server's Aldmeri Dominion.
Proud Founder of the Yaysay cult! DOWN WITH THE NAYSAY CULT!! #ToxicRemedy