While you probably could set up your own affiliate program on your own website, you are much better off to do what Susan said and spend the time working on your shop and your external websites working to promote your shop.
Setting up your own affiliate program isn't as easy as you would think. You couldn't just piggyback off of CP and use their CJ merchant program, you'd have to open your own - if not with CJ and you wanted to participate in a network, one of the others. Each having their own expenses to set you up in their network.
Even if you went on your own without the networks to promote you by offering your site inside their tools, you have the issues of having your own shopping cart, creating a method of tracking affiliate clicks, tracking cleared sales for each affiliate and writing checks to them. Doing all this, when are you going to have time to design and do the things that brought you to CP in the first place.

Now I'm gonna answer your questions in a different way just in case you weren't implying that you wanted to run your own program. CJ will issue more than one identifying number, one for each website you define in your settings in your CJ account. However, using PIDs from YOUR CJ account to connect to YOUR shop is against the CP affiliate program terms.
Instead of going through all the headache you are talking about going through, why not just help the people you recruit sign up at CJ. They could then provide you with their PIDs so you could apply them in the websites they run or you could teach them how to build the links in their sites. If they don't want to manage their CJ account, I believe they could even set you up as a user on their account so you could manage everything except where the checks go.