Hi Kenzi013,
We see you're running into some issues with read receipts in Messages, and we'd be happy to assist with that.
Just to clarify, when a user enables read receipts on their end, this won't change anything visually for that person in Messages. Rather, the person they're messaging should see the time stamp of when it was read. We note that you mentioned the one person notes it's working on their end, which would mean that if they're seeing the read stamp, that it's working on the sender's end.
If you're not seeing any time stamps on your side, either the person sending you messages hasn't enabled it, or it's not working for them. Settings > Messages > Read Receipts is one place to check for this. However, there's a separate toggle for individual conversations. Head to the Messages app > Select the conversation > Tap on the person's name at the top > Info. In here will be an option for 'Send Read Receipts'. This would also need to be enabled for the other person to see the time stamp.
If these don't help, be sure to also ensure the iPhone is updated at Settings > General > Software Update. If you or the other person need to update, be sure to back up ahead of time.
Take care!