You say you have a network. When people are using the network, how are they logging on?
Say you have Mac A and Mac B. Say Mac A's user is Joe and Mac B's user is Ann.
If Ann logs onto Mac A using Ann's username and password, then files Ann makes will be read-only to Joe, and Joe's files will be read-only to Ann.
Either Ann has to log on to Mac A with Joe's username and password, or you need to create a user (Ann) on Joe's computer that has read/write privileges.