The first referee is primarily responsible for net faults of the attacking team while the second referee is primarily responsible for net faults of the blocking team.
A player touching the net during a playing action, giving his team an advantage or interfering with the play are deemed net faults.
In detail these are the following actions:
Players may touch the net posts and the tensioning ropes as long as they do not interfere with the play.
It is not a net fault if the net touches a player due to a ball being played into the net. It is also not a fault if the hair of a player touches the net, unless it interefres with the opponent's play or a rally is interrupted (e.g. getting stuck with a ponytail).