They are actually deleted, unlike many other platforms. If not, the problem described here would not arise in the first place: https://blog.discord.com/how-discord-stores-billions-of-mess...
But I thought Discord, somewhat controversially, did actually let one person delete messages from all parties. Maybe they eventually caved because it enabled abuses/harassers to deprive their victims of evidence.
Although administrators can take away your right to even see the existence of a channel, which means that you no longer can delete your messages in that channel, since you have to see the messages to be able to delete them. For the record: if you have the permissions, you can delete anyone's messages in a channel.
Similar issues can arise in direct messages, for example if you closed the DM, and say, you left all mutual servers, and you unfriended your friend (without blocking), and the person changed his or her name so you are no longer able to add him or her back. When you add someone, that person will appear in "Pending" friends after having added, so you can click on "Message" to open the DM. Without that, you cannot open the DM, which again means that you cannot see the messages, thus you cannot delete them. I put "without blocking" in parentheses, because if you block someone, they will appear in "Blocked" friends, and it keeps track of changes in their name or even ID (that they are able to change with Discord Nitro), and you can open a DM easily from there. To open up the DM, you right click on the name, then click on "Message" which you can only do if 1) the person is in the same server you are, 2) the person is your friend, 3) the person is in "Pending" friends, or 4) if the person is in "Blocked" friends.
Alternatively, if you lose access to your account through losing your password, getting your account disabled, or removed by you or Discord. In this case, too, you lose the ability to delete your messages.
To bulk delete your messages, you can use https://github.com/victornpb/deleteDiscordMessages. If you have sent 100k messages, then it could take hours for your messages to get deleted due to rate-limiting. This script does nothing more than what you can already do manually; delete the messages you can see.
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TL;DR: You can delete (and edit) your own messages on Discord as long as you can see them (have access to them), and you can delete other people's messages from channels if you have the right permissions.
In some minority of cases, though, they would. Abusive ex sending you messages over discord? Yeah, those might be evidence especially when both parties are saying different things. This is the most common example I can think of, but I'm sure there are others. Heck, even if someone wanted to reminisce about days gone by... they should have the right to do so in conversations they were involved in.