"The fish is gone, I tell you."
Translation:Fisken er jo væk.
"Jo" is and interjection, and is used to emphasize or confirm the matter at hand. Kind of the German "Doch". And "I tell you" to "jo" is okay, though it's limited and very cultural/contextual and I think a correct answer should also be the more direct one like "Fortæller jeg dig". Actually the best, for me (as native danish speaker) should be "Fisken er væk, fortæller jeg dig jo", that is what you are saying with the shortened one - why I think both should be accepted for the purpose of understanding, when learning.
Question: Looking at the German equivalent: "Der Fisch ist doch weg!" implies an answer to a question or statement about the fish. Like someone asked: "Shall we have the fish tonight?" - "But the fish is gone!"
Appending "I tell you" makes only sense if there is some sort of repetition, like someone didn't get the fact that the fish is gone before.
Now back to Danish: Is it similar to the German use of "doch", or is it an entirely different construction?
Not entirely different. But still different uses. Less frequent than doch, in german. Also, "jo" is an affirmation as well, loosely examplified with "yes, we can do that" = "jo, det kan vi godt". So, mainly i'd use "jo" as another use of "yes" but also as and conjunction. Like in "The more I do that" = "Jo mere jeg gør det". Even worse [read: complicated] is "nå"... you could look it up in a dictionary but its so very cultural... with so many possible uses, mostly defined by the situation and intonation, it could be both a sound made for someone/something cute or when you feel offended by something. Its quite weird actually.
Check out this infographic
I think jo is really used to indicate a sort of contradiction, Skal vi ikke spise fisk i aften? Nej, fisken er jo væk. Also as a variant of ja: Skal du ikke hjem nu? Jo, lige om lidt, used after a sentence with ikke. German uses doch here, too.
This was really hard for me to learn when I first started with Danish. All of these "modalities were hard!
Because "jo" is an adverb. You could replace it with other adverbs:
Fisken er ikke væk = The fish is not gone
Fisken er helt væk = The fish is absolutely gone
Fisken er næsten væk = The fish is almost gone