[Do you demand proof from everyone or just those that think differently than you?]
I demand it of myself as well though not always.  This is mostly in cases where I believe it is common knowledge.  This is probably a waste of time but here are some of the reasons why I ask for a source.
- Legal/Academic standard: Self explanatory.
- Basic reasoning: Anyone can spout of a completely baseless claim.  Without a source one is free to disregard it.  Refusal to provide a source only lends one to believe it a fabrication.  Trying to shift the burden of proof only more so.
- Intellectual Honesty:  I recognize I am not omniscient so I have to provide an opportunity to provide a base to their claim.  
- Simple curiosity: Self explanatory.
- Interest in encouraging more substantive discussion: Self-explanatory.