Unscrew the hose from the left faucet and see if the water flows out of the right tank. If it doesn't, as Sally suggested, the hose is clogged. Screw it onto a regular hose and blast whatever's in the hose out.
Unless there is a filter on the rain downspout, the bottoms of the barrels are going to collect debris. Have you turned on the faucet on the right barrel?
