as channa correctly posted. When it comes to checking in at an airport what matters is the name on the outside of the plane. so if the plane is a UA plane then a person needs to checkin by UA, going to US or CO will only end up with being told to go over to UA

as for the bags. the CO CC allows for 1 or more free bags when flying 'on' CO. that means it says Continental on the outside of the plane. Otherwise you arent flying on CO and that means Unfortunately no free bag/s. the CC doesnt cover a code share flight as the Carrier whose plane is being used has no way to wavier the . In the OPs case even thou the UA CC is also issued by Chase Free bags isnt 1 of its benefits

I think this should be noted on CO.com in their baggage rules and exemptions as well as by Chase so that people would be properly informed