Does the meaning of "talk to" and "talk with" relate to their one-side or two-side conversation?
Skyminge is correct. Talk to implies a one-sided conversation, and "talk to a stone" is an excellent example of this. It doesn't have to be one sided though, for example "I talked to my boss today", and you'd assume that he talked back too. Talk with implies both people are doing some of the talking. For example "I had a good talk with my wife last night, though it felt like I was talking to a brick wall".
You are more likely to encounter talk with in American English. I have only heard it being used by Americans, on American television programmes and films. In the UK, people normally just say talk to. It seems that the differences between these are in their use. You can use talk with if you speak American English, but otherwise talk to will be enough.
It's the same with the words speak with.