Yes, the moisture is transferred to your sleeping bag. But unless you are sleeping in a plastic bag, it keeps going and makes its way out of the sleeping bag into the air because sleeping bag shells are porous.
By the way, there is no reason not to do this inside a down bag. Even if you sleep naked, your breath and body produce plenty of moisture. Your socks and gloves aren't going to make a huge difference.
Whether you use a down or synthetic bag, the moisture accumulation makes your bag less efficient every night you use it. This is probably more pronounced with a down bag, but probably not by much. My solution is to let my bag air out during the day (or for as long as possible between the time I wake up and break camp). It won't make your bag 100% dry, but it will do a pretty good job.
Winter air is really dry, after all, so it sucks moisture out of your bag pretty quickly.
Edit: There seems to be some misconception about how warm you will be with synthetic materials versus down. First, I know you can do this with down because I dry my gloves and socks inside my down bag every night when I am winter camping. I've been doing it for years in temperatures as low as -30ºF.
Second, while synthetic materials do retain some insulation value when wet, a 0F bag doesn't stay a 0F bag when it's damp. Try soaking a Thinsulate jacket, wringing it out, and then walking outside with it. Not warm, is it?
The point is, you want to avoid getting your sleeping bag soaked no matter what it is made of. But a little extra moisture from your gloves and socks isn't going to make any significant difference, especially since airing out your bag during the day is a pretty effective way to dry it out.