The Chat by @Siropu is 100% on your server; you own the data. But it does not have AV and may have a negative impact on your server's performance, if it is heavily used.
Discord, on the other hand, is a free community, were you can create your space (they call it "server") and with this addon by @NixFifty you can sync your forum's user accounts with their discord accounts. They have to create their own discord accounts tough first (additional to an account on your forum). Downside: You don't own the data on your Discord "server"; on your forum you may be an admin, but on Discord you are only a user, who runs a "server".