Flash memory multiple device I/O operations
The problem is how to control what gets done to the memory. For example, let's say both computers tried to overwrite the same file with differing contents - what should the result be? And they both will inevitably try it - if not any one file, then certainly the filesystem data.

If you want to access the same storage device from more than one computer, you can share it over the network (using nfs,smb,ftp or something else that shares files). Then the file server would be responsible for modifying the filesystem data and also for allowing or not allowing access to files (it may not allow write access to a file if it is open on another computer).

There is dual port RAM available (and it would probably be possible to make a dual port flash), but it is used when one port is used exclusively for reading, whiel the other could be used for reading or writing. And example would be the frame buffer of a video card - the processor reads/writes the memory while the DAC just reads it, converts to analog, and drives the VGA output).