How exactly are checklists shared between the pilots?
Typically, both pilots will set up "their" stuff as prescribed in the airline SOP's, and then the checklist will be called for.

At my airline, the captain has certain switches that he sets and certain systems that he checks -- generally things on his side of the overhead panel along with his flight instruments -- and the FO does likewise.  The FO performs the initial setup of the FMC, with both pilots checking what has been loaded (since some of the data comes in over a datalink).  Then, the captain briefs certain items, the clearance and routing are discussed, and then the captain calls for the checklist.  In flight, the flying pilot will call for the checklist (although if he's late in calling for it, the monitoring pilot my suggest, "are you ready for the Descent Checklist").

At my airline, the FO (or, in flight, the monitoring pilot) reads each item on the checklist, both pilots verify the switch or condition, and the captain responds.  Everything in the checklist is standardized, so the responses are predictable.  For instance, if the Before Takeoff checklist challenge is "Anti-ice" then the response might be "Off" or "Engine On, Wing Off" (since you don't operate on the ground with Wing anti-ice turned on in the 737).  But both pilots should be looking at the switch position to confirm that what is actually set (a) makes sense, given the conditions, and (b) matches what the Captain states.

Other operations will have some variation in this.  In another organization, the challenge "Altimeters" was answered "30.03, Pilot" followed by "30.03 Copilot" and then "30.03 Navigator".  Where I am now, we'd simply have both pilot check that the altimeters all match the ATIS setting, and the captain responds "30.03, set".  Either method works in a 2-pilot aircraft; with a Navigator and a Flight Engineer and a Loadmaster, crews in my old aircraft did things a little differently, in ways that made good sense in that operation.

So yes, the precise details of how all this is done are very aircraft- and airline-specific.