I\'ve made a simple test with a MemoryMappedFile as msdn says :
2 processes, 1 memory mapped file :
EDIT
I noticed one apparently strange thing in the code of ProcessB. This code
using (MemoryMappedViewStream stream = mmf.CreateViewStream(1, 0))
creates a view from the first byte, but the strings in .NET are 2 bytes. I think it should be enough to you to make that 1->2 become 2. So the offset of the ProcessB view from the start of the mapped file will be after already inserted "1" string from ProcessA.
In your case seems that you overlap them.
Hope this helps.