The patients' medical records contains the option to refer them to a specific player (from level 4 onwards).
After having clicked on the respective icon, enter the name of the player (please mind spaces and spelling in general) you want to refer the patient to, as well as the desired price. Once the transaction has been confirmed, the patient will wait in your waiting area or (if available) in your lounge, and will stay there until he/she is being accepted by your trading-partner or rejected and readmitted by yourself.
There are no points for already completed treatments when referring patients. Also note, that patients can only be taken over, if you are able to treat at least one of their diseases with your current level.
If a patient is referred to you, a red exclamation mark will blink at the card index on the top menu bar. You will be able to look at your referred patients' medical records there, accept or decline them. Clicking the magnifying glass will show you additional information about the patients, such as how many points have already been claimed by the referrer (check Patient Exchange).
Accepted patients will arrive in your waiting area/lounge, declined or withdrawn patients in one of the sender's sickbeds. That's why there might be partially treated patients in your waiting area/lounge. It's not possible to refer your patients from one hospital ward (game world/server) to another, you can only trade with players, who are registered on your ward.
You will find out if one of your patients has been accepted by a respective notification in the system messages.