Dog waste can be composted as long as the temperature of the compost pile becomes hot enough to kill any parasites. Generally, you should not add dog waste to regular compost bins or piles. There are composters on the market that are specially designed to handle dog waste. It is not recommended that you use dog waste compost in food gardens unless you are absolutely sure that any parasites and bacteria have been killed.
Cat waste should never be composted due to the risk of toxoplasmosis.
Do not put any pet waste in municipal compost piles unless you are specifically directed to do so.