Loyalty doesn't decrease the number of exercise wands needed if your base magic level is the same as your friends and if you're the same vocation. This is because a certain number of skill points are required for each skill. Loyalty gets added to the number of skill points you already have. When training, it's possible to lose and then gain a loyalty modifier after you increase whichever skill your training on. Related TibiaQA question: Loyalty Modifiers
I also went ahead and created a ticket for confirmation and this is the reply I received back:
"The needed amount of rods needed to reach the next basic magic level for 2 characters of the same vocation are always the same, no matter which loyalty bonus they have. Concerning training, the loyalty bonus doesn't give you any advantage. The loyalty bonus is always calculated on top of the basic skill. However, the basic skill is the same for everybody."
Q: Which skill is counted towards skilling? The one that presented at the website, without loyalty, or the one in game?
A: Interesting question and difficult to explain. Let's assume you and a friend have the same skill level listed in the highscores, e.g. 113. However, your friend has no loyalty bonus which means he has skill level 113 ingame while you have 118. You will both progress at the same speed and reach 114 at the same time (listed in the highscores). However, you might then still be 118 ingame since it requires more skill points to reach 119. CM reply