@cvollmann is right if your dedicated server is just storing your queue but if you wanting a dedicated server to run jobs then that is a little different.
You will need your codebase deployed to both servers. Your 'app' server will run like a normal website. Your 'processing' server will have Redis installed and your codebase deployed to it as well.
Configure both server's environments to point to the correct Redis server (as @cvollmann has said).
@TheFriendlyHacker yes on both servers but only one needs to be web accessible.
It would be good to use something like Laravel Envoyer to manage your deployments as you can deploy to multiple servers at the same time - super handy!
Also just to add to this, don't over complicate your app too early on. Managing multiple servers isn't that easy and if you don't NEED it yet then don't do it. You might find that one decent server can handle the jobs in the background just fine without the need to have a separate server.
Personally I have jobs for PDF processing which can be quite heavy but have never needed to have multiple servers for it - you'd be surprised at what these little VPS's can handle!
I want to clarify that I've read your question wrong/differently. Sorry!
If you would like to put only the queue to another server (as your title suggests) you don't need you application stack. But for "processing" your stuff @haganjones answer is of course absolutely right.
+1 on the suggestion not to overcomplicate things too early.