Different operators have different procedures. Who says what and when also may depend on how critical an item is perceived to be.
Typically before and after start checklists will be do-challenge-check-respond while many of the in-flight checks should allow one pilot (support pilot or PNF) to do-check-self-respond so as not to distract the pilot who is flying (PF).
Many operators have a double check of the landing gear before landing only e.g. PF calls 'gear down, landing checklist' PNF selects the gear and does the other stuff, like arming speedbrakes etc. Then PNF reads and self responds until he gets to 'Gear' which is a challenge. PF is required to check and respond e.g. 'Down, 3 greens' and PNF follows up with 'down 3 greens'. The reasoning behind this, is that at times of high workload, sometimes responses can tend to be automatically cued by the challenge rather than resultant from actually checking the item.
To prove this on checkrides sometimes I would start challenging from a totally inappropriate but familiar checklist and listen to the other guy give the responses to that checklist and not the one needed. Good for making a point.