At first: It doesn't matter if you use WinForms or any other template. You can easily add missing references to any type of project.
The two possible sockets would be either TCP or UDP.
A problem with these Sockets is that they are usually used for p2p connections.
Additionally only TCP guarantees the correct arrival of all sent packets, so in your case it would be highly recommended.
In this thread they discuss the problematic of many clients to one server:
TCP server with multiple Clients
If you made it to connect with multiple connections you should think of a system to handle each clients changes.
I'd propose a Server/Client system, where one person hosts the session.
Each client has a copy of the image and everytime someone changes a pixel a local event gets called that transmits the changes to the server.
The server now only has to "flush" the changes to every client connected.
One tricky thing could be two clients changing the same pixels at the same time, but thats your job =P