They don't, you only have to pay german taxes if you live in Germany, but you have to pay social security as you are based in Germany. And just to mention that, a lot of people complain that you have to pay more on social security in Germany, which is true, but you also get a lot more out of it, this is what most people forget.
@Reveal, Leipzig is a very nice place, a young city, lots of students. If you don't have a "home" to move back to, Leipzig is a very good choice.