First of all - welcome to BC !
I believe - or at least I hope - your description of the problem is not quite accurate ! I have set up many computers with my own on and using the same network without ever swapping files between one and the other. This allows me to do something useful on my computer while the other is, for example, downloading and installing windows updates. More to the point I cannot imagine a mechanism to allow this to happen accidentally.
Is it possible that you mean the desktop is identifying itself as 'Laptop' and the laptop is identifying itself as 'Desktop' ? If it is, it is reasonably easy to change the computers' names.
If the situation is as you describe then you are going to need an external drive to move the files to and then transfer back to the computers they should be on.
If you are using Firefox, to back up and copy your bookmarks, click 'Bookmarks', click 'Show all bookmarks' then in the middle top of the box that jumps out you will see 'Import and backup' , click this and choose 'Export bookmarks to HTML' and follow the remaining prompts to choose a file location where you can find it. Importing them is the reverse of this business.
In IE, select 'File' then 'Import and Export'. At the next screen choose 'Export to a file' and follow the prompts until you are asked where to save the file. Again choose somewhere where you can find. Open it in the other computer and all the bookmarks/favourites will be back where they should be. If you don't see a 'File' command in IE RIGHT click on any of the gray part of the top of the screen and click on 'Menu bar'.
Post back if you need more information or help.