I think the title is pretty self-explanatory. I am very happy with my current research topic but I would like to have some side research. Is it OK? And in case it is, how do people get those projects?
It's perfectly ok. In fact, I'd say it's a good idea to work on things that don't directly contribute to your thesis: you learn new things, you gain some valuable independence from your supervisor and your mind and creativity can be refreshed by tackling a new problem. Just don't let it become your priority, or I guess your supervisor might be unhappy.
People start side projects on their own, based on their own ideas and interests, or by meeting others and having interesting conversations. This can be as simple as chatting to someone at lunch in your institute, in the bar after work or at workshops and conferences.