So, I was poking around this forum for a different purpose, but this conversation induced a visceral need to respond, because I love the subject matter...
I *think* what the question being asked is how a congenitally (or very early) deaf person interacts with his/her inner monologue. That voice that composes and organizes your thoughts, though not necessarily in a grammatically correct language.
I've had the experience of learning a grand total of 5 different languages (or at least starting to learn and forgetting most of them): English from birth, Dutch and Thai between ages 3 and 7, French in High School (I actually remember enough to speak it), and Italian for a brief time before and during a month-long trip to Italy. Adding in ASL now (finishing up my first semester of it) and I have started to learn 6 languages, which sounds really impressive until I remember I can barely speak complete sentences in French, and have next to nothing to show for the rest. I can say "Hello" in Thai!
I may be somewhat unusual as an American, because I learned languages so young, but as soon as I start to learn a language, my internal monologue entirely shifts into that language. Which is why I always have to have a translation dictionary or I get driven crazy by incomplete thoughts
The experience is somewhat different learning ASL, because without closing my eyes it's almost impossible to visualize what the signs would be, and since I spend a lot of my time driving now trying to think in sign (which also induces my hands to MAKE the signs) it can be a bit awkward. If you see some guy driving down the road staring into nothing and fingerspelling "pneumonoultramicroscopicsiliconvolcanoconiosis," it's ok, I'm just practicing. Instead, what I'm "hearing" in my head is actually the sensation of making the signs, ie the expressive, but not receptive side (which I've always had trouble with in all non-English languages, since I've never been fluent). I would assume most hearing people, as I do, are actually hearing the sounds more than remembering what it feels like to move air through your larynx and modulate the tones.
The next train of thought this sends me down is... How do you think in English if you're a signing deaf person? Are you seeing the visual words in your brain, or running a translation algorithm from ASL or PSE to written English? And (in my boundless curiousity): How would an oral deaf person run his/her inner monologue? I don't know much about speech-training, but I'd assume it's every bit as consciously physical an experience as learning ASL is, since sound isn't the feedback mechanism in your body.
At this point, I'm doubting anyone read this far, so I'll stop musing...