It happens that I work in the space, at least tangentially. The company I work for is an audio conference service that basically invented the ability to deliver simultaneous interpretation over-the-phone. This was done at the request of a UN agency, who wanted to hold conference calls in the same manner that they held meetings in New York and Geneva. The UN is mandated to do everything in their 8 official languages.

With respect to streaming, I would think that you might encode multiple versions of the stream, each with a different language. That means a different URL for each language. A web client could have some logic to select language, but it's basically changing URLs. So really quite simple.

There may be other approaches, like making supplemental languages available as an audio stream, separate from the video. That implies a lot more intelligence in the client, and pretty much assures that A/V sync will be slippery.

In truth, A/V sync is not that big a deal since the process of interpretation itself means that the other languages are typically a bit behind. Sometimes just couple seconds. Other times, and with different language combinations, they can be many seconds behind.

In some languages the object/verb order is reversed, for example English vs German. As a result, the interpreter will listen to an entire phrase or sentence in German before being able to say the English equivalent.

As an example of how this is done, you might look at VoiceBoxer. They deliver multilingual webinars. Here's an example. It's just the english channel. When a non-English speaker is presenting you hear the interpreter.