I am a research scientist and I would like to know if I can do my own research at the institute where I am working? Own research meaning stuff that has to do with science. I have no legal papers signed on it or stuff like that. It’s just trying out new ideas that may lead to a project or a publication.
This really does depend on the institution where you're working. For instance, Google used to grant its employees 20% of their time as time to explore new things that had nothing to do with their actual job duties. (This may or may not still be the case.) On the other hand, one of my previous employers required time be charged to specific projects down to individual quarter-hours. As you might imagine, there wouldn't be a lot of time for individual exploration under such a scheme!
So consult with your supervisors: there may or may not be arrangements available to let you pursue "extracurricular" interests.