The answer is "Anne and Jack had had a chat on Skype".
Step-by-step explanation:
To describe what somebody else said we can resort to direct or indirect speech.
When using direct speech, we just repeat the message with the exact words. For example, Mary said: "I bought an umbrella".
Instead, when using indirect or reported speech, we tell the message without quotation marks. For example, Mary said she had bought an umbrella.
In this case, we ad the auxiliary verb "had" because the original sentence is in the past, and to retell it, we have to move one step back.