The best conversations I have are usually with myself. I learn and understand tremendous amounts by taking both sides of an argument and discussing it. I'm usually fair and civil, but if I start taking cheap shots on one side, the other side swiftly retaliates. By the end of the conversation, I either know which side is the winner or I understand why neither side can truly win.
There is a stigma associated with talking to yourself. It makes you look like a crazy person, somebody who hears voices. Well, in a sense, we all hear voices. The thoughts that pop into your head or creep up on you, that make you wonder about things, those are voices. You may not hear them through your ears, but we all recognize them.
The difference between going back and forth with thoughts in your head and doing so by verbalizing them is negligible. People will notice you either way. It will look strange. The only difference is that I've found verbalizing to be more productive.
When I hear my thoughts spoken, it is more like somebody else is speaking them. This makes it easier to consider them and argue with them. If the thoughts are all in my head, they swish and swirl around and I end up going in circles. When I make it a dialogue, it forces a natural progression that demands progress. And really, it's hard to argue with progress.
If you've never done it, give it a shot. If you feel nervous or embarrassed, just relax. When you start thinking, let those thoughts travel to your mouth. Let your body unwind and you will start speaking (or at least mumbling). You never know what you may learn from it.