dc, 35 bytes
?[zRla+salFx]sU[lb+sbz0<U]dsFxlblaf Input is a space-separated list of numbers on stdin.
Output is on stdout, on two lines: the sum of the left part, then the sum of the right part.
How it works:
? Read the input and push it on the stack. (The last number in the list is at the top of the stack.) [ Start a macro. zR Move the bottom item on the stack to the top, moving the rest of the stack down one item. la+sa Pop the top item on the stack and add it to register a. (This number comes from the left part.) lFx Execute F recursively. ]sU End the macro and name it U. [ Start a macro. lb+sb Pop the top item on the stack and add it to register b. (This number comes from the right part.) z0< If there's anything still on the stack, U then execute macro U (which adds the bottom item on the stack to register a and then executes F recursively). ]dsFx End the macro, name it F, and execute it. lblaf Print a, then print b.