How are you "Opening" the database?
Once the file is open on one machine (with FileMaker Network sharing turned on), you need to use the Open Remote to connect to it. NEVER, NEVER open the same file in 2 places by double clicking the file from a network drive, it can cause serious data corruption.
As for the limits of editing:
Look at the Accounts and Privileges. You should have different Accounts setup for each user. Then it won't really matter which machine the person is on, their login credentials will determine if they can edit data or not. It is possible to do what you are asking, but it is typically not a best practice.
Thanks for getting back to me - I actually created an executable (EXE) file for the databases. Does it mean that in order for the 2nd machine to share the files, I cannot open it as an EXE but via filemaker pro???
Correct. Run-times are not shareable.