The basic elements of teamwork: one is individual responsibility, which is always attached to another aspect that is positive interdependence. Individual responsibility is what you said before, that everyone must respond responsibly to their rightful part of the work within the team.
Another element is positive interdependence. This is, perhaps, more important than the last one. It is the feeling of belonging to a group. All team members have to have the following feeling: if the group fails, I fail. Both will have to occur, in a coordinated manner, for a team to work efficiently and successfully.
Obviously, there must be an interaction between team members associated with these two feelings: communication, there must be continuous and direct interactions between the different members of the team... and that team members apply different strategies, teamwork skills, which must be known and implemented within the team.