When you send a picture via iMessage on the iPhone or computer and ask that person to delete it, is there a way to know they have? If I delete the picture on my iPhone, will it be deleted on their iPhone?
No, you won't get to know if or when they delete a sent picture. Also, the picture sent is retained on Apple's servers in that person's receive queue until their space gets cleared over time and/or volume of new messages and attachments.
The only way you would know they deleted it was by observation, trust or having access to their account to check. You don't get any delete receipt from the system should they comply with your request. Lastly, deleting a message on one of their devices may not even delete it on others, so you might ask them to delete a message on all their devices that can receive iMessage attachments and monitor new devices for a period of time.