Sharing my unpublished thesis with other students?

Of course you should share your thesis: the purpose of doing research is to increase and spread knowledge, not to keep it secret.

Is it common to keep your thesis between you and your advisor - and potential researchers who may want to work with you?

No, it's not, and I suggest you to ask your advisor what he thinks of such an idea.

And now he wants to do research also and try and join a research group, etc.

Well, this is a good thing, isn't it?

By all means put your thesis out there. The whole point of academia is to advance knowledge within your field [and outside it] and that's not going to happen by people keeping their hard work under lock and key. Just a few days ago someone emailed their thesis to everyone in a mailing list I am part of, and very well received it was, too.

I know the temptation is to keep things to ourselves sometimes for fear of plagiarism or retention of originality, but this is not what academia is about. If you share your work you will affect and influence others by it, and you will get much further than if you didn't. The only time you should not share your work with all and sundry is if you are bound under a publishing agreement.

You should share your thesis. By your logic you should not publish either, since sharing your results through publications may reveal plans for future or current work.

Presumably you currently have expertise that others don't have, so you are ahead of any competition. It is hard to get results noticed, and sharing your thesis will increase the exposure of your results and future work. Moreover, sharing your thesis with fellow students will likely lead to clearer thinking on your part as students usually ask questions of a more detailed nature than experts.