Talking out of the side of your mouth
This means one is lying, right? Or something else?
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It means trying to keep a heated or private conversation private in a place where there's a risk that it may be heard, as Robusto suggested. Check this out: http://www.youtube.com/watch?v=cUKJWsnAAXs&feature=related&t=1m03s (one minute and 3 seconds).
To "talk out of both sides of (your) mouth" is to tell different parties different things. This would be akin to lying.
To try to maintain contradictory positions or beliefs in an attempt to please the most people. "We all know that you've been talking out of both sides of your mouth about the merger, so please, just tell us the truth.Will there be layoffs or not?"
"I don't trust that candidate—he still hasn't committed to a clear course of action and is always talking out of both sides of his mouth."