No. What I mean is, can you create a file on the desktop PC inside the Public folder, from the laptop? The Public folder is already shared by default. So if you can't write a file there, you have some other network issue. That's why I say it's a good idea to use that as a test. If you can't copy or create a file there from another machine, you won't get other folders not under Public to work, because there's some network setting or service that's not right.
I have Vista and Windows Seven. Between those 2 once I get access to Public folders the rest seems to solve quickly. Between those and XP(I have a dual boot PC) is more difficult.
But the starting point is to be able to create a file in Public on one machine, from the other machine in both directions. Once you get that then you probably have your homegroup or workgroup set up correctly.
When Microsoft did Vista they change the workgroup stuff around. For one thing the default workgroup name in XP was MSHOME. They changed it to WORKGROUP. Also it's counter-intuitive because they make Public networks require a password while Private networks can be accessed without one. You would think it would be the other way around but they mean Public as in publicly accessible, like in a kiosk, vs private, machines only on your home LAN.
First thing is make sure your workgroup name is the same on both machines. Use a Private workgroup. Disable firewall. Turn off av programs. Also make sure Network Discovery service is running.
I'm not a network expert. I get mine going by trial and error. Maybe someone more skilled can take you from here. But get write access to the Public folder on each machine from the other or else there's not much point in messing around. That's the easiest. Then you can tinker with mapping a folder to drive letter etc..