First, I'd make sure your rear turn signals are connected properly. Can't think of another reason ATM why LEDs wouldn't be lighting for you in such a system.
However, what's likely causing both sides to light is insufficient resistance in the system, since LEDs use so much less power than the OEM incandescent bulbs.
To correct this you need to do the "diode mod," which prevents the extra power from lighting the wrong side.
Two easy ways to do this:
buy a kit
or
DIY (pretty easy if you can solder, looks harder than it really is)