Share Automatically
After you set up name and photo sharing, choose to share your name and photo with only your contacts or have Messages prompt you before your name and photo are shared in a new conversation.
Keep messages
Choose how long to keep message conversations in your account. For example, if you choose 30 Days, your conversations (and all attachments) are available in Messages for 30 days. After that, they are automatically removed.
See Change iMessage preferences in Messages on Mac