Is it better to learn a language you are semi-fluent in or start a new language?
It depends on the reason. If you're suddenly moving to China and your Chinese is pretty good, then yes, it is better to learn the language you are semi-fluent at, that being Chinese, because you're going to need it. As for fun, I think it just depends on the usage or which you're more interested in. Obviously fluency will help, but you first need to know what you prefer. Or just do them all at once, but I don't recommend that for beginners because it can get overwhelming and you'll just end up quitting every language you were focusing on, or at least the majority of them.
Duolingo is not likely to be able teach you much in a language you are already 'semi-fluent' in, unless, perhaps, you are entirely unaccustomed to writing it.
Fluency is probably good. No problem with doing several languages at a time but it's best that you can hold quite a complex conversation before you finish a language.
Which alternative will likely be the most productive in terms of how much and how effectively you are able to learn, I think can be assumed.
Well productivity depends on what there goal is. People are not just using duolingo for amount
It depends. In my case, in the languages I am somehow fluent, duolingo (I suppose you mean learning a language here) serves mostly as a review tool of what I already knew, learning just a few new things. If you go from the very start with a new language (and as you can see I'm doing it with quite a few), the process will probably be slow, specially if you are not supporting that language with other sources.
I would do both things (I already do them XD) but the decision is yours.
With Duolingo you don't have to have any previous knowledge of a language to start. You can only get to an intermediate/advanced beginner level anyway.
But there's a lot of grammar and it helps if you are "semi-fluent" and want a refresher. Though if you are good at understanding, but not good at producing a language doing a reverse tree (learning English from that language) might help more, as you'll find more exercises where you'd have to translate English to that other language.
I'm doing both: starting two new languages (Catalan and Guarani) for Spanish speakers now that I've finished the Spanish for English speakers and English for Spanish speakers courses. :)
Once you're semi-fluent in Turkish, you can try the courses for Turkish speakers: https://www.duolingo.com/courses/tr . :)