I am a jeweler. No, it should not move--if it is rattling, it is loose and not seated well. You should have it put in a new head (the prong part) and if it is over .50 carat, it should be in a 6 prong setting. The 6 prongs will ensure that even if one breaks or is loose, the other 5 will hold it better than only 3.
As far as tightening it yourself, well, diamonds are brittle. If you tighten it too much or your tools move, you could very easily chip your stone. It is not as hard to do as one might think.
Third, and my own personal soap box, stone switching is absolutely NOT COMMON. In the jewelry industry you can purchase supplies and inventory from few places, and your credit is based upon a system not unlike credit scores. If you are stupid enough to switch a stone, you lose all credit, all your reputation, and essentially, your entire business. Now, there are stupid people out there. But not usually so stupid as to risk their entire livelihood on one stone--which there is squat for markups on diamonds anyway--there are so many more ways to make better money than switching stones. So, please, oh please, take that with a huge grain of salt. Especially when you are dealing with a stone under 2 carats--they are not going to waste their time on smaller stones.
Good luck, and get that head replaced.
A.