I have one slave site that I want to replicate.

This slave site is in it's own database (not shared with the master site).

I would like to replicate the slave site so that each site has it's own database (jos_)

Also, I want to do this as an Admin, and don't need to give front end access to create the new site.

What is the best way to do this?

You just create a JMS template that use your slave as "template site".
Once you have the JMS replate in which you have defined your replication (duplication) rule, you can return in JMS Manage site to create a new slave site using this replication rule.
You can also defined in your JMS template the default rule for the new DB replication (if you have the MySQL root permission).
When the new DB is located on the same MySQL host and that you have the MySQL root login that you have specified in the "mlutisites.cfg.php" file (see user manual) then JMS can also create the DB for you and also the MySQL user.

When you are not familiar with JMS setup, you can of course create the new DB manually with your cPanel, Plesh or any administrator tools.
Once you have the new DB parameters, you can provide them in the JMS Manage site (only when you have selected a JMS template rule).

I must not have the template parameters built right, then.

When I create a new slave site from the template, it puts the slave database inside the database of the template site.

Could you send a screen shot of how the template parameters should be entered to do what you're suggesting.

Which JMS version are you using ?
You are in the forum JMS 1.2.x and in JMS 1.2.x, you have the DB parameters

In JMS 1.1.x, you only have the table prefix

See video in tutorial step 16 that show the new screen with JMS template.
/tutorial

What I'm seeing in the tutorial is that all slave sites created from the template are in the same database with a different database prefix.

I wanted to create all slave sites in separate databases. If it's best to put them all in the same database, I will try that.

What is the best way to approach it?

No you can put them in another DB.
Your question was to know in which field you have to put the parameters.
I have replied to have a look in the tutorial to show you the field because I don't know if you are using the JMS 1.2.x or JMS 1.1.x.

Your question was not clear as let suppose that the fields for the DB was not present.

So now I suppose you have seen the fields and you can fill them.
When replicating into another DB if you don't have the MySQL root login, perhaps you should have to create the DB with you cPanel, Plesk, .... (admin tool) before setting information in JMS template.