Duolingo records how many continuous days you have done your minimum number of exercises (the standard is 20 points, or 2 set of exercises, per day, but you can set your own goal). It keeps track of this to motivate you, and you also get lingots (the in-game currency) depending on your streak. For 10 days, 1 lingot. For 20 days, 2 lingots and so on. Not sure if there is an upper limit to that. Not that I need lingots for anything anyway, but I appreciate the attempt at gamification.
I am currently on a 54 day streak, so the website says. Except I am 95% sure I missed Turkish one evening a couple days ago, when I was planning to do it on my tablet later so I did not have to change keyboard language on my smartphone. Various things happened and suddenly it was past midnight. The same thing happened with French a couple weeks ago, as far as I remember. But evidently Duolingo remembers this differently, because it seems to have forgiven and forgotten both of these infractions.
So my current theory is that if you are studying two languages, the streak is combined. If you remember do do at least one of them, the streak will be unbroken. This fits with the fact that I only got one set of lingots on round days (5 lingots for 50 days, not 10, not 6 for one 50 day streak and one 10 day streak). In other words, streaks are for people, not for languages. Or that is my tentative conclusion so far.