Step 1: Look at silver's black tarnish first.
Silver hardly reacts with oxygen or moisture, but there is one gas in the air, released from small amounts of pollution and organic matter, that it does react with, hydrogen sulphide, $H_2S$. This reaction coats the surface with black silver sulphide. \[ 2Ag + H_2S \to Ag_2S\ (\text{black}) + H_2 \]
Step 2: Now look at copper's green coating.
Copper reacts slowly with a combination of moisture, oxygen and carbon dioxide in the air, building up a green layer of basic copper carbonate over time. \[ 2Cu + H_2O + CO_2 + O_2 \to CuCO_3\cdot Cu(OH)_2\ (\text{green}) \] The carbon dioxide is the key extra ingredient here compared to silver's tarnishing.
Step 3: Compare the two cases side by side.
Silver's blackening needs $H_2S$, copper's greening needs $CO_2$ along with moisture and oxygen, they are two entirely different atmospheric culprits for two different metals.
Step 4: Conclude.
The gases responsible, respectively, are $H_2S$ for silver and $CO_2$ for copper. \[ \boxed{H_2S\ \text{and}\ CO_2} \]