If you have a V778, you are entitled to sell (or give) the associated registration number to another person. However, the DVLA advises against sharing scans or photographs of your V778, as someone other than the intended recipient could use it to put the private number on another vehicle.
If you are selling (or gifting) a registration to someone, you should use the V778 document to assign the registration to their car. This can be done online (via the DVLA website) or by post.
The DVLA will then send a replacement V5C logbook to the registered keeper of the vehicle, containing their new private registration number.