Here's a breakdown of the implied meaning:
* Saying different things to different people: This person might say one thing to please someone and then contradict themselves when talking to another.
* Being insincere: Their words may not align with their true intentions or beliefs.
* Trying to manipulate or deceive: They may use different arguments to get what they want from different people.
Example: "Don't trust what she says. She always talks out of both sides of her mouth."
In essence, this idiom suggests a lack of consistency and integrity in someone's words and actions.