First of all, it's important to remember that budgies are opposite to mammals. Males have two Z chromosomes, and females have one Z and one W (as opposed to human males who have one X and one Y, and human females who have two X chromosomes).
In budgies, W chromosomes have no connection to mutations and what the feathers will look like, all they do is make a budgie female (to put it simply). Therefore, female budgies can only carry one copy of a sex-linked gene on their one Z chromosome, whereas males can have two copies, one on each of their Z chromosomes.
Females cannot be split for sex-linked mutations, and they cannot pass them on to their daughters, because any chick that gets the Z chromosome from it's mother will be male, and any that gets the Y will be female, and unable to carry a sex-linked mutation on that chromosome.
When figuring out what you will get when breeding sex-linked mutations, it is obviously important to take into consideration the sex of the parents. For the purpose of this I will use Opaline, but it works the same for all sex-linked mutations.
Male: Normal/Opaline, Female: Normal
Male chicks: 50% Normal, 50% Normal/Opaline
Female chicks: 50% Normal, 50% Opaline
Male: Normal/Opaline, Female: Opaline
Male chicks: 50% Opaline, 50% Normal/Opaline
Female chicks: 50% Opaline, 50% Normal
Male: Normal, Female: Opaline
Male chicks:100% Normal/Opaline
Female chicks: 100% Normal
Male: Opaline, Female: Normal
Male chicks: 100% Normal/Opaline
Female chicks: 100% Opaline