Does this sentence imply that he is just starting to study Esperanto for the first time? or that he has already been studying for some time and is just starting to study right now.
Or has neither implication and depends on context?
What are peoples thoughts on the following as a variation? "he begins the study of esperanto"
Any feedback on how/why/where it doesn't work?
It means the same, but is syntactically quite different. You can try reporting it, but I'd still advice to try to stick closer to the original sentences (in that case, using the verb "to study" or "studying").
I am not a native English speaker, but I'd say that in English is far more common to say that you learn a language than that you study a language. To study, in English, has more to do with academic studies, so if you say that you study Spanish, for instance, you are more likely to be following academic studies on Spanish at university, rather than learning by yourself or going to a language school. So, even though it is obvious that studi and study share a common steem, I think that "He is starting to learn Esperanto" should be consider a valid answer, since it actually is the more natural in English.