How much time on my clock do I have to lose before it is legal to stop notating? When I play in tournaments, I do not know when it is legal to stop chess notation. Also, if my opponent stops notating, is it okay for me to stop notating as well?
According to the FIDE Laws of Chess:


  Article 8: The recording of the moves
  
  8.4 If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then for the remainder of the period he is not obliged to meet the requirements of Article 8.1. 


Article 8.1 is the one that states that players need to keep record of the moves and what the players must and must not write to the scoresheet.

So:


  When I play in tournaments, I do not know when it is legal to stop chess notation. 


You may stop notating when your clock shows less than five minutes and you get no increment or increment of less than 30 seconds. You have to start again at the start of the next period, that is, when more time is added to your clock (not the increment but for example at move 40).


  Also, if my opponent stops notating, is it okay for me to stop notating as well?


It makes no difference whether your opponent has to write the moves or not.