The fishing rights seem to be owned by the Eibsee Hotel. Here is whats found on their website. 

http://www.eibsee-hotel.de/ : 

Fishing is a real pleasure here. And one that will capture your senses: Experience the serene atmosphere in the mornings being among the first to take your boat out on lake Eibsee. Watch the crystal-clear lake with the reflection of the Zugspitze massif while nature is slowly awaking at sunrise. This unique setting will let you get away from the stresses of daily life.

The lake Eibsee belongs to our family and our guests enjoy exclusive use of the lake for fishing. During the fishing season from 01 May to 01 October, mainly lake trout, rainbow trout, vendace, tench and other kinds of whitefish are caught.

You only need to bring your fishing license to obtain a fishing permit for the lake Eibsee from reception.

Once back from your fishing trip, our kitchen team will be happy to prepare your freshly caught fish for you. Just let us know how you would like it prepared and savor this freshest of all dishes in our restaurant!

So, do not forget to bring you own fishing equipment!

Wiki: 

Eibsee is a lake in Bavaria, Germany, 9 km southwest of Garmisch-Partenkirchen and roughly 100 km southwest of Munich. At an elevation of 973.28 m, its surface area is 177.4 ha. Eibsee lies at the base of the Zugspitze (2950 meters above sea level), Germany’s highest mountain.

As of May 2010, trains from Munich to Garmisch-Partenkirchen leave roughly once an hour and the trip takes about an hour and a half.[1] Buses travel regularly from the train station in Garmisch-Partenkirchen to Eibsee.[2] A valid train ticket can often be used to ride the bus.

https://en.wikipedia.org/wiki/Eibsee