Can horizontal stabilizer trim be worked independently (each side)?
No.  Theoretically you could have the left and right stab move independently so there is still some trim authority if one jams, but would be structurally way more complex and quite heavy, your trim authority would still be cut in half, and there isn't really a strong need to do that from a risk perspective.  There may be aircraft out there that are like that, but most moving stabs are single surfaces usually operated by a big electric jack screw.  

The required redundancy is provided by redundant control channels for the trim system, dual drive motors for the screw jack that drives the stab, and a dual load path design in the acme thread and the trunnion (the "nut" part of the jack), and in the attachments that connect the trunnion to the stab and the actuator assembly to the structure.  The only single point of failure would be for the hinge of the stab itself to completely seize, and the risk of that is low enough that it doesn't need to be accounted for in the system architecture.

Two independent stab surfaces would require an attachment to the fuselage or fin that allows independent movement of each surface while cantilevered off the root structure, which would require a pretty heavy and complex center structure for not a lot of gain safety wise.