It is alright to include students' first names in a teaching statement? Is it alright to include students' first names in a teaching statement, or would this come off as weird and too personal? Or a privacy concern?  

You often hear the advice to include specific stories that exemplify some of the things you claim about yourself in your teaching statement. I've got an anecdote involving two students I want to tell. My thought is that including their names will demonstrate that this wasn't just some frivolous incident that I'm talking up; that I honestly got to know these students. Also, I think the story would read more clearly if I can give them names rather than pronouns.
I think you should avoid giving their full names, as opposed to just their first names unless they give permission for it. Better, using an alias for the students protects their privacy but doesn't cost you any thing in the telling of the stories. Alice and Bob, as usual. 

But some readers might actually wonder whether you were too personal with the students, not keeping a proper "academic distance". I often tell a story about two students who succeeded after failure. It is a great story, and it works without names. If they heard my story they would recognize themselves, I think. But no one else would. Not even their classmates at the time.