Within the search menu it would show an arrow pointing (diagonally) down & left for recieved or (diagonally) up & right for sent beside the message. It would also display the name of the person who sent it under the chat name (for groups). Additionally, you would have the option to search through specific chats or all chats.

This is my idea to make searching for messages easier because it can be a bit confusing as of right now. Thank you for taking the time to read this.

-Kal