Google Messages now lets you control who sees your profile picture and name through the Profile discovery feature. Users can choose from three privacy settings: everyone they message, only ...

This is part of “Profile discovery,” which appears in Messages Settings > Advanced once rolled out to your phone. It is a Google Account-level setting that you can turn on/off.

When starting a chat on Google Messages, Profile Discovery will default to using the contact's Google Account picture. This would have been fine as an option, but users weren't given the option to ...