If there's something "off" about manual control of either the lights or relay in your Double Switch, or if they don't work manually at all, then something about the installation can probably be improved.

Review the diagrams in the manual and if you're unable to locate the source of the problem, take some pictures the switch box according to the angle and instructions shown here. Then, send us the images so we can have a look and see if we can make any recommendations. 

Have a look at the air gap switch underneath the relay button and check that it's pushed in.  This switch is a safety feature that cuts power to the device when you're changing bulbs. When pushed out, power is disabled and this may mimic a dead device. If pushed in, power should be enabled. 

Another thing to check for is that local control of either the lights or relay wasn't accidentally disabled; this is especially relevant if you can control the Double Switch fine with your Z-Wave app but can't control it manually. You can enable or disable local control by tapping the lower paddle or relay button 4 times and holding it on the fourth press. Try this out to see if it restores local control.

Also, check that the dimmer and relay are connected to compatible loads. The dimmer portion of the Double Switch should really only be connected to incandescent or LED bulbs within the recommended loads (so that means no tube lights, halogens, DC powered fixtures, or chandeliers). The device connected to the relay part of the Double Switch should not exceed 15A or 1/2 HP. 

If you need more assistance with the troubleshooting process or suspect that the type of load you've connected has damaged the Double Switch, let us know!