Vixjean: It was very intimidating the first few times I began lifting in the free weights section as well. It helped me to have a staff member demonstrate several exercises for me that rounded out a fbwo when I began. After a while I became more comfortable around people who would lift during the same mornings as me and began to add new moves to my routine. It did take a lot of courage for me in the beginning though. It has been a very empowering experience for me overall and I'm so glad I have added this to my life.
Nelie gave a great suggestion. If you have time to do it on an off hour you don't have to worry about anyone really looking over your shoulder or running into any "regulars" from your normal workout session. I have also found that making your space in front of the mirror gets much easier the more often you do this. I know I don't have to tell you this but... you pay the same fees for membership as everyone else not to mention that you have every right to take care of your body in the way that is best for you. You can play in the "sandbox" too!
It might also help to spend some time researching some exercises you may want to do with the free weights before you head over to the gym. Research the moves online, maybe practice their implementations at home without weights, write it down in a small notebook and take this information with you to the gym. A notebook or piece of paper will help remind you of the exercises you want to hit so that you don't go blank once you sit down on the bench with the dumbbells. It will also be a record of your strength gains in reps/sets/or weight you are using from the very beginning to the many months from now when you are an experienced beginner.
Good luck and happy lifting.