What! keep a dog and bark myself?

The speaker sees no point in taking over work that someone else has been given to do. The dog's bark stands for the task expected of the one being kept.

The rhetorical question can justify leaving a responsibility with the person assigned to it, rather than doing that person's work as well.

Use it when someone suggests that a person should perform a task already assigned to another person. It gives a pointed, humorous reason for refusing to take over, but may sound dismissive if directed at the person responsible for the work.