~$(then the name I used to save it.)
Those files may just be for system use. When you use your program to open one of your files for editing/viewing, a temporary "lock" file associated with the file you've opened gets created, preventing other network users from modifying the same file while you're doing your work. When you close the file the temporary lock file is supposed to get deleted, but there are times it doesn't (program crashes, improper shutdowns, network disconnects, etc.).
If Explorer's View Options are set to display hidden files you'll see these files get created/deleted as you open/close your data files. This setting allows you to see all other sorts of files as well, and may help to explain what you're now seeing.
If another user tries to open a file you've already got opened for editing, it'll let that other user open it, but in a read-only mode.