XD on desktop without other programs?

To get the release version of XD you would need a CC subscription of some type. If you have the free subscription you can download a trial of XD that would last 7 days. You could also subscribe to XD as a single app for $9.99 per month. Here's the different plan types:

Creative Cloud pricing and membership plans | Adobe Creative Cloud

If you already are a subscriber at work, and you sign in under the same Adobe ID at home, you can use XD and any other app you are subscribed to.

The rules about how many computers you can install to are described here:

On how many computers can I use my Creative Cloud subscription?

As it says on that page, the short version is:

"Your individual Creative Cloud license lets you install your apps on more than one computer, activate (sign-in) on two, but use them on only one computer at a time."

That would mean you can remain signed in on your work and home computers, but if you're at home using XD, nobody can be using CC apps on your work computer at the same time.