Is there a clear-cut definition of which genders hi/shi/zi/etc apply to? I don't know any (outwardly) transgender persons and thus don't have any opportunity to use these pronouns, but even if I did I would stick with "they" for the risk of misapplying one of these other pronouns. Is there one that strictly applies to nonbinary/third-gender persons? One that strictly applies to agender persons? One that strictly applies to transgender men, or one that strictly applies to transgender women? I want to be progressive in my vocabulary, but I just don't understand it well enough.
My understanding is that gender neutral pronouns such as zi/zer (which are the two I'm told to use the most but there are quite a few) are just that. They can be used for anybody, and all it means is that you are not specifying a gender.
For transgendered people, if you're worried about pronoun use and not sure about gender neutral pronouns, the best rule is to go go with their gender and not their sex. For example, my older brother is ftm, so even though he looks female and has a lot of feminine qualities, it's still best to refer to him as he. But gender neutral pronouns are really useful when you have people who's gender is neither male or female. And you'll find people out there who really dislike being placed in either gender.
I'm unaware of any third gender pronouns that are to be used just for people who are neither male or female, but I'm really no expert on this either. If I ever hear of one, I'll post it in this thread.
Fun fact: English actually used to have gender neutral pronouns back in the middle ages. Ou and a.